Before you even begin reading this build, it's important to realize that I am by no means that talented of a LoL player, and I'm not intending to make a tried and true, ultimate, competitive build. I'm a pretty casual LoL player, as far as League players go. I'm just posting my build to give more people the chance to get into Anivia, since she's very underplayed (as far as I've seen), and can easily carry late game. Unlike a few champs (*cough* Tryndamere *cough*), Anivia isn't really OP'd by anyone's standards. If you're doing well with Anivia, you probably deserved it. This build is very different by many players' standards: first of all, I put most of my points in the Utility tree, instead of Offense. Second, I get far less magic penetration than most Anivia players (none in the core build). I hope that you take this guide for what it is, and don't just tell me how much better I could be doing, because overall, this build works pretty well. I appreciate thoughtful, constructive criticism.

I used to take an Armor & AP Rune page (+21 Armor, +26 AP), which is nice early game for really most AP champs, especially Anivia. However, I've been playing against quite a few people recently who have Greater Glyph of Scaling Magic Resist, and I'm barely doing anything to them mid game, so I tried replacing my Greater Glyph of Ability Power with those, to help more with defense. I ended up using a combo of the two. They're very useful runes, even for Anivia.

Anivia's main stat is AP, as all of her skills scale with it. Also, Anivia is extremely vulnerable to ganks, especially at lower levels when you don't have completed boots, and higher leveled skills to slow/stun/block your attackers. Anivia is both one of, if not the, squishiest and slowest champ in the game. One of her early game saving graces against this is her passive. Armor is important in general, especially for Anivia, who has low armor, and can easily be crushed by AD-heavy champs if not careful. The extra armor'll help you survive longer. The AP, again, is self-explanatory.

I'm not a fan of Greater Glyph of Scaling Ability Power, but if you're fond of them, you could have any combination of those and Greater Glyph of Ability Power.

Explanations
Greater Mark of Armor: Extra armor is always nice, especially on Anivia, who has very low armor. Since you don't really need magic pen unless the enemy builds MR, I prefer to get the extra defense over a little magic pen. You get about as much armor from these runes as you would magic pen from Greater Mark of Magic Penetration.
Greater Mark of Magic Penetration: I've recently changed my marks to these, in favor of some more damage. I was building Anivia a bit too tanky for her role, so I've decided to change.

Greater Seal of Armor: These're probably the best seals in the game, and armor's always nice, especially since Anivia has very low armor. Since there aren't many other seals worth using, (mana regen runes're jokes) much less for Anivia, I suggest taking the tremendous armor boost.

Greater Glyph of Ability Power: I really believe that offensive runes're for the early game, and these give you some more damage. If you prefer, you could go for Greater Glyph of Scaling Ability Power, but as I said, I prefer the early damage. Late game damage is gonna come from items.
Greater Glyph of Scaling Magic Resist: I've noticed quite a few people lately using these glyphs, and I've barely been doing anything to them mid game, so I decided to follow suit. In 5s, you're almost guaranteed to fight an AP champ, so it'll help with defense. In 3s, you may end up laning Tryndamower or Darius, so they could potentially end up not helping as much, but they'll still help once you get out of the laning phase.

Greater Quintessence of Ability Power: These runes're much better than Greater Quintessence of Scaling Ability Power. These give you AP. AP increases your damage. Damage makes you hurt. You wanna hurt as this champ. Simple.
Greater Quintessence of Movement Speed: When using Sorcerer's Shoes, Anivia is simply too slow to secure most kills, or escape from lethal combat. Using these quints makes those much easier while still getting the magic pen from the boots.

I build heavily in the Utility tree, as opposed to the Offense tree. The simple reason for this is that the VAST majority of your AP/damage is gonna come from items anyway, and the extra damage granted from the Offense tree isn't enough of a lure for me. The lack of damage from the Offense tree is easily filled by items, while many of the bonuses granted from the Utility tree simply can't be filled with items that suit Anivia. I was initially very turned off by Strength of Spirit , but once I thought about it, I realized that that's one of the best Masteries a mage can get. Anivia gets over 3500 mana with this build, which adds at least +35 Health Regen, without building anything else for it. Since Anivia is so squishy, this helps you survive (assuming that your low HP doesn't get you killed in one fight anyway).

Explanations
Mental Force : With runes/masteries, I go for the early game damage, and this'll top your AP off to 30 from the start.

Sorcery: Combined with Intelligence , your passive CDR'll be 10%, which is pretty nice, especially since I don't get any CDR items on Anivia.

Summoner's Insight : Since I take Clarity, and Clarity helps a lot more with this mastery, I take it. Since I now also use Flash for Anivia instead of Ghost, it's a no brainer (having both Summoner Spells in the same improvement mastery and all). If you don't use Clarity or Flash, you should remove this mastery for one point of Good Hands .

Expanded Mind : Anivia is a very mana hungry champ, and this gives you extra mana. Again, how in-depth do you need me to go? I'm assuming that you aren't that inept.

Swiftness : Anivia is an extremely slow champ, and this gives you a little boost to MS. Since you need points to get deeper in the Utility tree, this is a nice choice. Greed doesn't help much at all, and spell vamp doesn't provide much for Anivia (especially a mere 3%), as most of her damage comes from AoE/multi-target spells, so Transmutation isn't very alluring. Again, since you need more Utility points to get deeper, go ahead and get the extra MS.

Meditation : As you read above (if you're reading at all), Anivia is a very mana hungry champ. This gives you extra mana regen, and you still need more points to get deeper in the tree. Get these masteries to help with some extra mana.

Runic Affinity : Since you're most likely going to be mid (in 5s), you're probably gonna get blue (buff) at one point or another. Since blue is perfect for Anivia, and you want its benefits for as long as possible, and this gives you exactly that, you should take this mastery. Also, this is the final key to unlocking the bottom tiers of the tree. Even in 3s, the MS, CD, and damage buffs from the top jungle are very valuable, and this allows you to keep them longer.

Awareness : You're finally getting into the bottom of the tree. This mastery's really gonna help you get the level advantage over your opponent(s). That makes things easier. Yet again, most of these masteries're self-explanatory.

Strength of Spirit : You're finally there, the reason you built in the Utility tree. This is an excellent mastery for any mage who uses mana. 1% may not sound like much (I know it didn't to me at first), however, consider you have 3500 (which isn't unreasonable to get). This mastery adds +35 health regen (near a fully stacked Warmog's ( Warmog's Armor)) without building anything for it. On a mage, this can be very nice. When coupled with some actual health from RoA and Rylai's, and armor and MR from runes, you actually have some endurance, even as an otherwise squishy mage. Since most of your damage is gonna come from items anyway, and having to build for this while using the Offense tree, would severely take away from your build. If you just ignore the health regen, your tank(s)'d better be very good at redirecting incoming damage for you, or you'll have to B back to base after you receive any damage. Since you're gonna be soloing for a large portion of the game, I highly suggest taking this mastery to help you stay in the lane for as long as possible. Also, Expanded Mind helps get the ball rolling on this mastery even quicker.

Intelligence : More CDR can't hurt, and even 2% more CDR is more useful than Mastermind . If you really like Mastermind, feel free to switch out one point of Intelligence for it, otherwise, stick to this.

I used to have Summoner's Wrath , but I'm gonna start trying Flash instead of Ghost for Anivia, so that mastery'd be pointless.

The main portion of the build. Items are the single biggest game changers. The items which I've listed are what I consider to be the core items, and you must get regardless of what the other team looks like. After those, you can build situational items, such as Zhonya's Hourglass if the enemy team is AD-heavy, or Abyssal Mask and/or Void Staff if their magic resist is really getting in the way. If you honestly don't need any situational items (very unlikely), I suggest Rylai's Crystal Scepter and another Archangel's Staff (AAS).

EDIT: I used to get a Rabadon's Deathcap (Deathcap) before finishing the Tear of the Goddess into an AAS, but I just decided that the AAS is worth getting before a Deathcap even if it only has 400-500 bonus mana or so.

EDIT: I'm gonna start getting RoA before TotG for some early-mid game damage, which is when this build is at its worst. You still wanna get that TotG as early as possible to start building early mana, but I'm really, really weak getting Catalyst the Protector, Tear of the Goddess, then all the damage items. I still love my end game, I'm just changing the order of the early items. Just be more aggressive about building the TotG/AAS at the shop.

About the boots: Anivia is extremely slow, and the little amount (yes, little amount) of magic pen granted from Sorcerer's Shoes doesn't hold a candle to the massive speed boost via Boots of Swiftness. These boots make it much easier to chase enemies who may be slightly faster than you when your skills are just out of range, or for escaping ganks. Also, I've outplayed every Anivia I've seen (granted, I haven't seen that many), when every single one takes Sorcerer's Shoes.

Building Tear of the Goddess/AAS: Anivia is a very easy champ to increase the bonus mana from TotG/AAS with. A. Most of the regular fighting you'll be doing will work towards it since you need your spells to do basically any damage, B. Clearing the lanes is extremely easy with Anivia thanks to Glacial Storm. Since you'll be using this spell quite often as means of an extremely effective farm, you can count on getting a substantial amount of bonus mana just from earning gold.
C. You get to double dip with this and Flash Frost by hitting Q before the spell reaches the end of its range, or by simply toggling Glacial Storm off. Both the activation and the cancelling of both of these spells contributes towards building the item, AND both spells have pretty quick CDs.
D. While healing/at the shop, launch off a couple of spells utilizing this method to get some more mana. Don't be afraid to use your wall in the base when you first get back, either. It may have a long CD, but by the time to get back in the lane, it'll be refreshed.
E. Even though AAS doesn't truly shine until it gets towards the end of its bonus mana, don't be afraid to get it even if your Tear only has 400-500 bonus mana or so. The highly increased mana regen is very nice. Doing so will cause you to become stronger just by simply fighting/casting spells. Your AP will slowly increase along with your mana, and by the time your AAS has 800+ bonus mana, you'll get an extra 100+ AP just from the item's passive.
F. It's important to remember that the bonus mana passive on AAS is unique, however, the 3% max mana -> AP is not. Purchasing subsequent A.A.S.s (not always recommended) will not only help even more with mana regen, but will give you the most AP out of any item (the highest AP one could get from items is one Deathcap and five A.A.S.s; Deathcaps should never be stacked under any circumstances). By buying a second AAS, you could get 150+ AP and another +25 mana regen. If your defenses/health aren't an issue, you should definitely consider getting a second AAS.

Situational Items
Anti-AD: Zhonya's Hourglass
Anti-AP: Abyssal Mask
Anti-MR: Abyssal Mask, Void Staff
Anti-CC: Moonflair Spellblade
Anti-Tank: Deathfire Grasp
General (if you still have space): Rylai's Crystal Scepter, Will of the Ancients, Morello's Evil Tome
Winning Too Hard to Care: Another Archangel's Staff

If you look at almost any build (and I mean this sincerely), seemingly regardless of the champion, the author'll tell you to take Flash and Ignite. While I can see why so many people go for Ignite, I truly, truly don't understand why so many people take Flash, and so few take Ghost. I see the allure of Flash, it's a very useful escape/chase tactic, however, if it doesn't single-handedly get you the kill/out of danger, you're screwed. Ghost may be a slower start-up, but it's much more stable.

I can only imagine how many people're cringing at the sight of Clarity. Most people consider it a worthless spell, or at the very most, not as useful as other Summoner Spells. I think that Clarity makes laning much easier early-mid game, until you get Tear of the Goddess/ Archangel's Staff, when mana becomes nearly no object.

EDIT: For Anivia, I've been thinking about how Flash could be used for devastating combos with Crystallize, so I don't have nearly as much hatred for Flash anymore, although, Ghost is still preferable on most champs, imo. Remember, if you remove Clarity and Flash, replace Summoner's Insight with one point of Good Hands .

Ghost: Anivia's extremely slow, and Ghost makes it much easier to escape/chase. I prefer this to Flash since Flash is really only useful on Anivia to jump your own Crystallize, and Ghost'll make securing your spells a hell of a lot easier.

Clarity: Although it falls off by the end of the game, Clarity does a fabulous job of replacing most mana potions throughout the entire course of the game which would actually require potions at all. Since Ignite is mainly useful for the Grievous Wound, and most mid champs aren't gonna beat you purely due to spell vamp, I prefer to focus on keeping myself in the lane.

Flash: I'm gonna start trying Flash out. Due to this, I'm removing Summoner's Wrath . Flash is useful for jumping your own Crystallize, as well as chasing/escaping. While it isn't as stable as Ghost, it has the added usefulness of jumping over terrain. Also, if you're pretty sure that you're gonna die, this could be used to better position yourself for Rebirth, such as within a tower's range. I'm hoping that the Boots of Swiftness'll be enough base MS to not miss Ghost too much, but we'll see.

I hope that if you've made it through this build with an open mind, you can at least see where I'm coming from. Thank you for reading, and I hope that this guide at least helps a few people pick Anivia up to make their own builds.