Welcome to my Veigar build. It is my first build and I'm more than happy to open up to constructive criticism (:. I have been playing LoL for almost one and a half year now. My first champion was Veigar and I instantly fell in love with him. I've played countless other AP champions, but none are quite like veigar. In my build, I will teach you how to deal insane burst damage with Veigar.

As Veigar is a burst champion, 21-0-9 is optimum for his damage output. the 9 in utility tree would allow stronger lane sustain ( Through bonus mana/lvl and mp5) as well as a tiny spellvamp. I put one point into the spellvamp mastery as even though it's only 1%, baleful strike is a single-target nuke which gives it full spellvamp. Over time, it would heal me for alot, allowing me to stay in lane for a longer period of time.

Runes

Rune pages are highly varied from different people. I like to go 9 Greater Mark of Magic Penetration because these, combined with your masteries, would deal nearly true damage to champions early game. 3 Greater Quintessence of Ability Power just for that little early game 'oomph' and easier farming with Baleful Strike. I take AP/lvl blues and yellows as Veigar don't need too much AP early game; rather, he's a late game nuker.

Even though I sacrificed a lot of utility (mp5, movespeed,), this would allow a higher damage output.

There are other rune pages and masteries which I would recommend.

Variation 1:
changing Greater Mark of Magic Penetration to greater mark of knowledge, followed by changing Sorcerer's Shoes to Ionian Boots of Lucidity. This would give you more AP at level 18, as well as a sweet 15% Cooldown reduction. However, this means sacrificing 30Mpen, which would be offset by buying Void Staff as soon as possible. This way, there would be no flat mpen to reduce the value obtained from %mpen. more CDR means also means more Baleful Strike per second.

Other changes:
Change Greater Quintessence of Ability Power to Greater Quintessence of Movement Speed and/or changing the 3 points in Meditation to Swiftness . This would be good if you're against long-ranged pokey champions like Ahri or Brand, where you need the mobility to dodge their skills. Or if you're against high mobility champions like Katarina to be able to chase up. Or just for that extra few movement speed to ensure Event Horizon lands.

This section is about Summoner Spells that should or should not be taken with Veigar and why.

Good Picks

Flash Flash - Your free get out of jail card. It has countless possibilities for Veigar. Enemy too far? Just flash over and stun-kill them. Enemy's hot on your tail? Just flash over a wall and laugh as they walk a big round around it. It is (in my opinion) a must pick.

Ignite Ignite - A pretty neat summoner spell, the only one dealing damage to champions. True damage. Just a 'lil nuke for early game to ensure the first blood. Another very fine pick

These two are what I would almost always pick on a Veigar.

Decent Picks

Teleport Teleport - Casting this on a friendly creep/turret/ward would teleport you there in 4 seconds. Quite a neat skill, useful in quite a lot of situations. Allow you to teleport to other lane to snagged a kill or two every few minutes; allow you to recall back to base, grab some items, and be back before your opponent even knew you were gone.

Heal Heal - A rather popular pick on many squishies after the buff. Quite sweet, giving you that few more HP which might save your butt. Not really recommended as Veigar isn't meant to tank a lot of damage.

Clarity Clarity - Heals some Mana. Would not really recommend this skill unless you are really horrid at managing mana (face-rolling on your keyboard 24/7 to ensure that all your skills are casted once cooldowned), otherwise his passive and the meki pendant would be more than enough.

Avoid

Pretty much everything else.

I will be explaining my choice of item here.

Start off with Meki Pendant and 2 Health Potion. Some may prefer to go Boots and 3 Health Potion, but I highly recommend the former for a very nice amount of mana regeneration. Pre-6, pretty much play passively and keep farming your Baleful Strike for a decent amount of AP bonus late game. Be sure to retain about 200~300 mana in case your jungler wants a gank, so you could drop your Event Horizon and Dark Matter and poke with Baleful Strike.

On the first recall, optimally you would be able to upgrade your Meki Pendant into a Tear of the Goddess, grab a Boots and a kage's lucky pick. Tear of the Goddess is quite an underrated item on Veigar; it gives a huge amount of bonus mana (from its unique passive) and more mana regen. Remember, more mana = more mana regen from passive. Constant farming of Baleful Strike would help to charge it pretty fast as well. Kage's lucky pick is grabbed early for it's gp10 function, getting it early would help to pay off its cost.

Basically, you would want to continuously farm your Baleful Strike whenever possible. Nearing level 6, you might want to start poking your opponent, bringing them to about 30%~40%. Once you hit level 6, combo them and dump your ulti and you should be able to net a kill. Once over level 6, try and gank whenever possible as it gives about 3~5ap per champion killed, not including gold bonus, exp bonus, and farm denial. When you can't gank, don't stupidly rot in a brush. Rather, go back to your lane and farm your Baleful Strike like no tomorrow.

You should rush a Deathfire Grasp and complete your Sorcerer's Shoes. Deathfire Grasp is a nice nuke addition to your combo; the damage it deals (if the opponent is full hp) can be somewhat around the damage your ulti deals. Sorcerer's Shoes is next as the movement speed bonus ensures that those fast-running balless opponents do not outrun you, and 20mpen would just helps you to get closer to dealing true damage.

By now, you should have a decent amount of AP, approx. 100++ . You should be going around more frequently to net kills. With Deathfire Grasp, place it at the first slot. now your combo should be E-W-1-Q-R (ensure that 1 lands before W to have maximum damage). You should be able to 1 shot any squishes without trouble. From all the gold you get from nuking people's *** and creeps, you should rush out your Rabadon's Deathcap, 140AP and 30% of your total AP is just a lot. with 100AP from your Q, it gives nearly 300AP in total. At this point in time, you should rake up to about 500~600AP in total. Here is where you decide; Lich Bane or Archangel's Staff? if your Tear of the Goddess's charge is less than 800, i highly recommend you going for Lich Bane first. With Lich Bane, your AP should approach 700 (maybe even 800!). This means in addition to your combo, you can auto-attack once at the end of the combo to deal extra 800 damage. How neat is that?
Another trick of Lich Bane is that it helps you to push like a maniac. as it can proc against towers, each spell cast followed by an autoattack on the tower would allow you to deal ~800dmg every 2 second, a very very strong backdooring tool.

When the opponent starts stacking Magic Resistance (MR), you can consider getting a Void Staff or Abyssal Mask. but the question is, Void Staff OR Abyssal Mask ? A little bit of quick calculation would be needed. Since there's 30 magic penetration from your boots and runes, you have to consider how much MR your opponent would have. the rule of magic resistance is this: Flat MR reduction > % MR reduction > Flat Mpen > % Mpen . In other words, if the opponent have 100 MR, and you opt for Void Staff, it will be 0.4*(100-30) Mpen, which is 28mpen. this will leave your opponent with 42 mpen. however, if you go for Abyssal Mask, it would be 100-20-30 Mr, which is 50 mr left for your opponent. But Abyssal Mask is an aura, which would benefit your team greatly if there's more AP carry other than you.

By now, when the opponent haven't surrendered or lost yet, finish up your built with Archangel's Staff for that insane mana to AP bonus. Your final could be anything. Getting focused? Up a Zhongya's hourglass. want to backdoor quickly? Up a Warmog's Armor just for that extra luls and 1350 HP. by now your damage should be so high, it doesn't really matter what you build anymore.

This section is dedicated to this item.

This item, Mejai's Soulstealer, is highly situational. Owning the opponent so badly until Baleful Strike and Dark Matter is enough to kill their tank? By all means go and grab Mejai's Soulstealer and snowball like nobody's business! It has the most AP bonus for a item, for the low low price of 1200gold. It gives (when fully stacked) 180AP.

On the other hand, if you're dumping Primordial Burst at your opponent's AP carry and it doesn't even tickle him, I highly suggest you do not grab Mejai's Soulstealer as it would be a TAD bit harder to stack it (: