Veigar is undeniably the most powerful single target nuke in the game. He is also one of the greatest carry in the game thanks to his insane scaling and permanent AP increase. Despite his lacking early game, you'll still be able to carry hard into late game. If you disagree with this statement, run as fast as you can, because I'll come get ya.

Anyways, Veigar is the ultimate killer of this game. Once I get my Rabadon's Deathcap, I can even OHKO tanks with my full burst, because Veigar can just dish out that much damage. However, despite his insane carry potential, he rarely ends up doing so due to his despicable laning phase in which he'll end up being denied, and probably even feeding.

Most guides place Pros vs Cons in a neat list, but I'll just quickly mention the basic to you. Pros? Best nuker in the game. Cons? Squishiest champion in the game.

There aren't many occasions in which Veigar can serve as a viable member of a team. And now I'll explain the generally rare situations in which Veigar can contribute to his team.

BUILD 1: Regular Carry Build (30-40 minute games)
BUILD 2: Hyper Carry Build (45+ minute games)

Veigar by all means should never, EVER solo a lane. Just no. Please don't do it. Think of the starving children in Africa.

I'm under the assumption that you're in a decent ELO playing against decently skilled players in ranked mode. Veigar is generally not a viable champion. He's never picked in tournaments. It's not because he's a mean yordle, it's just because he absolutely sucks in the laning phase. So in most situations, you should not be picking a Veigar.

Situations in Which Veigar is Viable:
1) Double AP Carry team composition
2) Viable AD Carry to solo at mid
3) Strong Jungler to damage gank
4) Pubstomping noobs/smurfs

Well why is Veigar so unviable? Because he will get denied in practically any lane. You can't even start off with a Doran's Ring due to your insane mana problems. Popular top lane champions ( Jax, Irelia, Xin Zhao) will just lunge at you which initially shaves off literally half of your HP, Exhaust you, and BAM YO' DEAD SON. Popular mid lane champions ( Annie, Anivia, Malzahar) will be able burst you at level 2 and ignite you to death.

His auto-attack range is an atrocious 525. His Q (which you must be farming with 24/7) is 650. Which isn't so bad, except the fact that he practically has no poke/harassment to even keep him safe. He can't harass with Q because he should be farming it. He can't harass with W because the enemy will just dodge it. He can harass with E + W, but that costs 180 mana at level 1. Your mana will be sent down the drain immediately, and you won't even be able to farm up your Q.

"But I have my passive mana regen!"
No. It gives you about 2 mana per second when you're at about 10% MP.

"But then I can raise Perserverence mastery"
No. It will give you about .2 mana per second during the laning phase.

Against any decent player, Veigar absolutely can NOT survive without a laning partner. And they can't be farm-dependent champions eithe; making Veigar such an unpopular pick. I know you've seen professional streams with Veigars destroying at mid lane. That's because the enemy they went up against were total derps.

Although still not the most viable pick, Veigar fares much better in normal games, where people aren't nerdy tryhards. Well, at least not as much. The key to surviving a solo lane in Normal as Veigar is to play passive. We all know playing passive is for pussycats. But what is this tiny little yordle supposed to do?

However, no matter how passive you play, there are certain champions designed to just destroy passive players. I will separate them into "Solo Mid Champions" and "Solo Top Champions."

Solo Mid Champions:
1) Caitlyn
2) LeBlanc
3) Fiddlesticks

There are many other champions with long range skill shots ( Lux, Brand, Anivia, Cassiopeia, and Malzahar). For these champions, I recommend that you start off with Boots + Potions. It'll make running away from their long range harass attacks much easier. However, the problems arise with LeBlanc and Fiddlesticks (who should be jungling, but lanesticks is so annoying). LeBlanc can just Sigil of Silence Q harass you all day. Fiddlesticks can just E harass you all day. Both of those skills outrange Veigar's Baleful Strike. Caitlyn can just harass you with her auto attacks because she's annoying like that. Her auto attack range is equal to Veigar's Baleful Strike.

When faced with champions like these, forget about winning the lane. Focus on not feeding, farming up your Q at least, and protecting your turret. You will be sacrificing health every time you walk up to Baleful strike a minion. Smart players will actually just stand in front of your minion wave and deny you completely, which will be a pain in the ***. Since you'll be under your turret constantly, keep in mind that two auto attacks + two turret attacks = dead warrior minions and two auto attacks + one turret attack = dead mage minions. Two auto attacks equal one Baleful Strike. Always remember this, and time wisely.

Solo Top Champions:
1) Jax
2) Irelia
3) Xin Zhao
4) Pantheon
5) Mordekaiser

Jax, Irelia, Xin Zhao, and Pantheon all have lunges that equal the range of your Baleful Strike. Not only that, three of them have stuns and one of them has a knock up. These are dangerous champions you want to completely avoid as if you innocently walk up to Baleful Strike a minion, they'll just jump on you, stun you, exhaust you, then kill you. And Mordekaiser just outranges you with his Siphon of Destruction, and with his Iron Man passive shield, he's impossible to harass. morde es #1 op huehuehue ok? Just avoid them.

Just farm. Farm. Farm. Don't imagine yourself as an evil yordle with insane capabilities of black magic. Imagine yourself as a hardworking farmer yordle. And that's what you'll do until you get your first damage item - Deathfire Grasp. By then you'll also probably be at least level 6, in which you'll be able to OHKO any champion.

Since you have Teleport, don't be afraid to port back to base if you're low on HP or something. During your first visit good items to grab are Doran's Ring(s) (I usually grab two before building my core items), Kage's Lucky Pick Kage's Pick, and Boots.

Once I port back to base for my Deathfire Grasp, I usually go gank. Combo that enemy Annie into a decomposing carcass of an innocent little girl and mangled pieces of a child's stuffed teddy bear. Is this to graphic for you? Then don't play a Veigar, because this is what you should be doing to every single champion out there. Nyehehehehe...

Woah. Sorry about that. Moving on. Anyways, after my first gank I usually just go back to farming at bottom. Also, buy a Sight Ward ward. I don't care what your name is, where you came from, or what your great ancestors did. Buy a damn Sight Ward ward. Do you ever watch tournament games? Every player buys Sight Ward wards. There are at least 5 Sight Ward wards on the map at all times. Map awareness is the key to victory. And if you still don't understand, hopefully this video will enlighten you. And it will give you a good laugh (skip to 2:56 for the important part)
Play Video
You should realize that the enemies WILL have Sight Ward wards of their own. It's crucial to be aware of the enemy team. Pay attention to when they're buying Sight Ward wards, what kind of items they're building. Ask for pings on lanes that purchased Sight Ward wards so you know where not to gank.

During team fights, you obviously should never be leading your team in the front. I also do not recommend kiting with Veigar, he is just far too squishy with no escape mechanisms. The enemies will just Flash on your face and laugh at your dead body.

You should also note that many champions are meant to bypass your tank/dps and charge in to your row of squishies and destroy each and every one of you. Especially YOU. You are the squishy of all squishies. You are the glass cannon of all glass cannons.

Once a team fight occurs and your noob teammates are just clashing at whoever's in front of their faces, it's your job to pretty much destroy the carries. What you want to do is either stand back or wait in a nearby brush. Wait for all the CCs to pass by ( Amumu, Warwick, Annie, Orianna, Sona, WHOEVER). After the initial CC spam, you just stroll in there and land an epic AoE stun. Most squishies don't even need your entire burst chain to kill, so you can split off your Deathfire Grasp and Primordial Burst to separately burst down two squishies. However, after your initial burst you are going to become their main target. And this is when you pop your Zhonya's Hourglass and laugh at their team while your companions demolish errybody.

Or if you're feeling super confident and you're quite fed, just destroy their initiator and leave their team helpless. If an Amumu Bandage Tosses your teammate and is about to cast his Ultimate, stop his bandage midway with your Event Horizon and just release your nuke and destroy him/her. Of course, this is under the assumption that you can either OHKO them or damage them to the point your DPS can just auto-attack 'em once or twice to their deaths.

You should also consider that decent enemies consider the same strategy. Smart Annies don't initiate, they wait to Flash in and destroy the squishies, especially due to her insane AoE spells. Many other champions (mainly tanky dps) are capable of bypassing the forefront and destroy squishies ( LeBlanc, Jax, Irelia, Fiddles, etc). So be careful for that. Always keep in check who's in a team fight and who isn't.

The greatest counters against Veigar are Quicksilver Sash and Banshee's Veil. If a squishy has a Quicksilver Sash, just stun them to make them waste it. Don't bother trying to use Dark Matter, just unleash your other three skills and they should be going down pretty easily anyways. If a tanky champion has Quicksilver Sash, just don't target them (unless they already used it) and you should be fine. And if they have a Banshee's Veil, just pop it with your Event Horizon (your cooldown is much shorter than theirs).

And finally, for the sake of Demacia and all that is holy, DO NOT MISS YOUR STUN. Trapping your enemies in the middle of Event Horizon is NOT what you should be doing. It might work when they're running away, but smart players with quick reflexes will immediately turn around and dance and laugh while you miss your W, or just Flash out. If you miss your stun, expect me to go over to your house personally and punch you in the face. Yes, I know where you live. Just don't miss your stun, okay?

'Nuff said.

Sorcerer's Shoes:
Pretty self-explanatory. But shoes are quite situational. You can opt for Mercury's Treads (counter champions with faceroll CCs such as Fiddlesticks or Taric) or Boots of Swiftness (counter champions with annoying pokes such as Brand or Caitlyn). Do NOT get Ionian Boots, because you'll just exceed your CDR cap with Masteries, Runes, and Deathfire Grasp. Total waste.

Deathfire Grasp:
This is the item that was MADE for Veigar. It destroys 30% of their current health. But you don't even need to initiate with it to get its full effect, making it such a great nuke item for Veigar. You can destroy any champion with this item. Even without it you will still have the strongest nuke out of all champions. However, this item gives you the potential to OHKO all champions mid-game and OHKO two champions at once during late-game.

Zhonya's Hourglass:
Explained previously. If you're successfully picking off squishies in team fights, you can opt for Void Staff first. However, this is an amazing defensive item, especially for epic glass cannon champions such as Veigar.

Abyssal Scepter:
Yes, I do realize that the Magic Pen on Abyssal Scepter has a diminishing value due to your Void Staff. However, in that logic, you might as well not get Sorcerer's Shoes either. Percent Magic Pen allows you to demolish tanks, but flat Magic Pen allows you to demolish squishies... Even more than you already should be doing. You should also keep in mind that enemies can have negative magic resist, which Void Staff does not allow you to accomplish. I would actually recommend getting this item BEFORE Void Staff if the enemy team isn't too heavily stacking on AP. I would recommend getting a Void Staff instead if the enemy has two champions with over a 100 MR. Also, I'm sort of OCD, so I like to balance out the extra Armor I get with Zhonya's with the extra MR I get from Abyssal. :P

Rod of Ages:
Normally I'd recommend this to any AP carry, as they're usually the first targets in any team fights. If I saw an Annie that didn't get a Rod of Ages, I'd call her a noob (unless her team was pubstomping and she rushed a Rabadon's Deathcap for a quick win). However, I wouldn't get this item on a Veigar because once he puts down his stun, he literally forms a barrier around himself. Quickly nuke someone down, and run. If someone Flashes over your Event Horizon to chase you, just Flash away. If you're in a situation where you're caught by enemy champions, the extra 600 health points (although it sounds nice) won't do anything for you. Health is nothing without armor/resistance. It works on Annie because she has a somewhat tanky skill with her Molten Shield. It works on Orianna because she has a somewhat tanky skill with her Command: Protect.

Lich Bane:
This would just force you with another attack in your burst combination, which is just really overdoing it at this point. Your stun will disappear before you even have the chance to use everything.

Rylai's Crystal Scepter:
This item is really not worth it. It gives nice health points (read Rod of Ages) and ability power, but the slow will do absolutely nothing for you. Just don't get it, okay? Again, think of the Africans.

Archangel's Staff:
I really, really, really don't like this item. This will put Veigar at such a disadvantage until super late game. You can get Tear of the Goddess first and leave Archangel off until your other base items, but that just means you're 1k gold further away from nuking down champions easily. Sure the mana points and regen helps. But you can just be conservative with your skills instead. Consider that if your game has gone far to the point in which your Archangel's Staff has become an advantage, you could have ended the game much earlier without wasting your gold on it. However, due to multiple complaints and suggestions, I have added this onto the SECOND build, the Hyper Carry Build, where you should expect the game to go on a long time (you should see this coming during the early laning phase when there's total stalemate).

Mejai's Soulstealer:
If you're facing a troll team or just an extremely unskilled team and you're getting fed, nab a Mejai's Soulstealer to snowball your little evil yordle butt to victory.

...

Are there any other situational items?

Items I Do Recommend:
1) Banshee's Veil: Generally a great item. MR and protects you from CC. Just make sure you don't aimlessly walk around and get your BV popped by a poke.
2) Quicksilver Sash: The most underrated item in the game. I would actually recommend this over Banshee's Veil against some champions ( Veigar, Taric, Malzahar, Fiddlesticks) as BVs are generally not that difficult to pop. However, feel free to get this item WHENEVER you are being targeted and have problems with CC.
3) Shurelya's Reverie: Maxes out your cdr without Elixir of Brilliance. Even so, it's still a great item. Especially if you're not starting out with Ghost.
4) Will of the Ancients: If you're doing exceptionally well with survivability, this item is a great item for the ballsy ones. With full cdr your Q is at about 2 second cooldown. And with all the awesome AP you farmed up, you should regain health in no time.

Items I Don't Recommend:
1) Warmog's Armor: No. Health won't save you. A ranged dps does about 1k per shot when their item set is full. There goes your Warmog's.
2) They'll just lifesteal your Thornmail away. Not to mention, YOU should be nuking down dps champions before they can kill you.
3) Pretty much, you're so unbelievably squishy that a few extra armor, mr, or health won't do much to protect you. What you need are items that prevent you from getting hit.

Equilibrium:
This is an outright atrocious passive. His old passive was great until Riot decided it was op and nerfed it to oblivion (like they do with everything). I did hear recently that Veigar is soon going to get either a rework or a revamp in his passive. So I won't say much about this for now.

Baleful Strike:
Farm this all day. Remember, you are a farmer yordle. Do NOT harass with this skill, ESPECIALLY during the early levels of this skill. It has an 8 second cooldown at level 1 and using it to harass only means extra AP you're going to miss out on.

Dark Matter:
Your main burst skill. Many Veigar players argue that you should max Baleful Strike first (so you can get two extra AP per champion kill). But honestly, that's just stupid. Dark Matter deals infinite more damage. I'd rather get a champion kill with 3 AP than a champion assist with a potential of 5 AP but only 0 AP instead. Or hey, that champion might as well get away.

Event Horizon:
Dat stun. Like I stressed before, make sure you hit the enemy champion with the edges. Otherwise they won't be stunned. Definitely practice this skill a lot. At level 1, Event Horizon stuns for 1.5 seconds and Dark Matter channels for 1.2 seconds. This means you need lightning quick reflexes if you want to land your burst. If you're having trouble, feel free to add a few more points to it. If I see that enemy champs are building Mercury's Treads, I raise Dark Matter and Event Horizon in turns until Event Horizon is 3, then I max Dark Matter.

Primordial Burst:
People dub this as the "Ultimate AP Killer" but it's not. It really isn't. It scales off enemy AP, yes. And 80% is quite a damage. But the enemy AP carry will be so squishy anyways that you won't even need the extra scaling to destroy them. However, this is an awesome nuke with some insane damage and scaling.

Nuking Sequence:
1) Even Horizon
2) Dark Matter
3) Deathfire Grasph
4) Primordial Burst
5) Baleful Strike

Yes, there will be times when you nuke that you realize you could have killed them with just Baleful Strike - thus saving your Primordial Burst. But in all honesty, it's really not that big of a deal. You'll have insane CDR and you'll get the skill back in less than a minute. And the guaranteed kill is great; saving you the bitter face contortion when your opponent gets away with 3.5 HP because you wanted to conserve your ult.

Flash and Teleport are my main skills. Flash by all means is a must-have spell.

Teleport however, I shall be more lenient with towards your preference. Teleport ganks are insanely evil, and I personally think there should always be at least one champion on the team with Teleport to prevent annoying backdoors.

Decent Alternatives:
1) Ghost: If you like to use your Flash offensively, Stun + Ghost provides great escape.
2) Clarity: At best it will give you the chance to harass a few times and last hit with Q a bit more. I personally don't think it's worth it. However, it will give you the chance to start off with a Doran's Ring.

Meh Alternatives:
1) Cleanse: If you're really having problems, get a Quicksilver Sash. But I suppose it has its uses.
2) Clairvoyance: I find Clairvoyance a great skill to have in a team. If your team has no support champion ( Sona, Janna, Alistar, etc.) with Clairvoyance and you feel that your team REALLY needs this skill, I suppose you can nab it. Awareness of enemy jungle and potential ganks is always nice.
3) Ignite: Your nuke should be able to OHKO your target. I suppose Ignite can help out during those rare occasions, but it's a skill better left for other teammates. Your skill sequence is already long enough.

"r u dum?" Alternatives:
1) Revive: r u dum?
2) Fortify: r u dum?
3) Smite: r u dum?
4) Rally: r u dum?
5) Heal: r u dum?

Just make sure you switch around your Masteries depending on which Summoner Spell you choose. I also explained that you should not be picking Perserverence over Good Hands. Unless you're in like a Co-op vs Ai game or something. Everything else is pretty self-explanatory.

Quints: Flat AP
It will help out with the farming early game. And those Q last hits are essential. Also, I completely do NOT EVER recommend AP / Level Quints because they simply are just not worth it. They even out with flat Quints at level 12. By level 18, it gives you a total of 8 AP bonus over flat AP runes. And that really doesn't mean anything AT ALL late game. It gives you about 25 extra damage on your full burst. I'd rather have 14 extra AP at level 1.

Glyphs: Flat CDR
CDR Glyphs are just great. However, cooldown reduction is always capped at 40% and at super late game when you finish your build and have a Morello's Evil Tome, your cooldown reduction will be at about 46% so it will be somewhat of a waste. Most games you will not reach this point, but if you feel really nitpicky about it, go for Flat AP runes. It will help with the early game, and you shouldn't go AP/Level for the reason I previously mentioned.

Marks: Magic Pen
There is no alternative for this. Do not opt for the AP Marks, they are not worth it.

Seals: Mana Reg / Lvl
Yes, I JUST argued against per level runes. However, seals are an exception because Veigar will have mana problems all throughout the game. And Mana Reg / Lvl Seals even out much earlier than most other per level runes (Level 6 to be exact). However, since Veigar's main mana problems are early game, if you really prefer Greater Seal of Replenishment Flat Mana Reg runes, go for it.

Just don't derp on your stun, and you'll do great. :)

Criticism and +1's will be greatly appreciated. <3
If you had the courage to read the whole thing, props to you brutha.