Before you even begin reading this build, it's important to realize that 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. Anivia is a champ that is very commonly played in ranked, but almost never in unranked. Although she can easily carry late game, Anivia isn't really considered to be OP'd by anyone's standards. If you're doing well with Anivia, you probably deserve it. I build Anivia pretty differently than most players, mainly for the fact that I build heavily in the Utility tree, as opposed to the Offense tree (I also didn't use to have any magic pen :P). I appreciate constructive criticism, however, don't waste your time commenting if all you're going to do is complain about how I'm not following the meta, or something equally pointless.

This is my previous Anivia build.

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. 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 on Anivia, but if you're fond of them, you could have any combination of those and Greater Glyph of Ability Power. I may end up changing to them now that TT doesn't begin with a team fight like it used to.

Explanations
Greater Mark of Magic Penetration: I used to use Greater Mark of Armor, since they give you about as much armor as these marks do magic pen, but that's building Anivia a bit too tanky for her role, so I switched my marks in favor of a higher damage output. I also used to use Boots of Swiftness as opposed to Sorcerer's Shoes. I've played Anivia in numerous ways in the past, so trust me when I say that magic pen is almost mandatory for her.

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: These're also useful runes, even for Anivia, but since I'm gonna start getting Athene's Unholy Grail (previously a worthless item for Anivia in 3s, imo), MR won't be much of a problem.

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. (NOTE: The following stats are based on my Anivia build for old 3s, but still prove my point) 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).

I've tried numerous mastery page set-ups for Anivia, from 0/9/21 and 0/8/22, to 21/0/9, and this is my current mastery page.

Explanations
Mental Force : This gives you a slight boost to your initial AP, which can help in the early game.

Sorcery: Combined with Intelligence , your passive CDR'll be 10%, which is solid.

Summoner's Insight : I would take this mastery just for improved Clarity, and since I now use Flash (I used to use Ghost instead) too, it's a no brainer.

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 : This honestly won't be as useful as it used to be in 3s, or is in 5s, since the top jungle buffs no longer exist, however, it'll still be useful for when your team gets Vilemaw's buff, and is the final key to unlocking the bottom tiers of the tree.

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

NOTE: Some of the stats and items for the next description reflect the old 3s build, however, still fit into the central idea of why I get the mastery.
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) without building anything for it. On a mage, this can be very nice. When coupled with some actual health from RoA and Rylai's, armor and MR from runes/masteries, 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 : CDR isn't extremely helpful for Anivia, however it's still more useful than Mastermind .

Stay tuned for major changes once season 3 begins.

The main portion of the build. Items are the single biggest game changers. Unlike my old 3s build, my new build isn't as tied to the item order. Feel free to mess with the order of some of the mid game items as you see fit for the situation.

Situational Items
As always, I suggest getting Rylai's Crystal Scepter if you have the space for it. You should get Rylai's unless the enemy team has two or more AP champs, and they're really hurting you, in which case, get Abyssal Mask. If for some reason you're dying mainly due to CC and not necessarily damage, then you could build Moonflair Spellblade, however, I wouldn't really recommend it. If the enemy team has a tank, and he's carrying, consider Void Staff, but otherwise, try to avoid it. 49 flat magic pen is more than enough in almost all cases. Aside from these four, I can't think of anything that you should spend your money on.

I'm including this section because it's important to me to see where a lot of my ideas for this build came from. I almost exclusively played 3s on TT before the remodel, so I know a thing or two about it. The biggest change between the two builds is the items. Anivia can now benefit from an extremely different item set, however, this doesn't mean that her old favorites are obsolete. If you're looking for some situational items, or ideas on how to play Anivia differently, this is a useful section to read. In fact, I'm sure that the old items would still work, however, one of the main things Riot wanted to do with the remodel is shorten the average game length of 3s, and a lot of my old items were based around the long, late game.

My old core items used to be as follows: Sorcerer's Shoes, Rod of Ages, Archangel's Staff, Rabadon's Deathcap. If things were going well, I'd end with a Rylai's Crystal Scepter, and another Archangel's Staff.

The rest of this section is segments pasted directly from my previous build, mainly dealing with reasonings behind my old items.

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.

This is what I consider to be situational in 5s and old 3s.

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 have space): Rylai's Crystal Scepter, Morello's Evil Tome
Winning Too Hard to Care: Another Archangel's Staff

Going 21/0/9 on Anivia also works very well, depending on your team. If you have two tanky AD champs as your teammates, you can go 21/0/9 on Anivia and easily carry most games (depending on the enemy champs).

Unique to Utility
Unique to Offense
+21 (more) Health Regen
+6% CDR (which hits a full 40% at full build)
+5% Exp.
+1% MS
Runic Affinity
Damage X1.015
+46.4 AP (at full build)
Executioner

You could do the extremely stereotypical Flash/ Ignite combo, however, I think that taking Clarity makes laning a lot easier. Since you'd only need Ignite for the grevious wound on Anivia, I don't think it's necessary. Ghost is one of the most underrated Summoner Spells, and, in fact, I used to use it in place of Flash for a long time. The only reasons that I starting using Flash instead is so that I could jump my own Crystallize, and 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.

OP'd: Overpowered
AD: Attack damage
AP: Ability power
MS: Movement speed
MR: Magic resist
CD: Cooldowns
CDR: Cooldown Reduction
AoE: Area of Effect

RoA: Rod of Ages
TotG/Tear: Tear of the Goddess
AAS: Archangel's Staff
Deathcap: Rabadon's Deathcap
Rylai's: Rylai's Crystal Scepter
Warmog's: Warmog's Armor

TT: Twisted Treeline
SR: Summoner's Rift
3s: 3vs3; Twisted Treeline
5s: 5vs5; Summoner's Rift
Old 3s: TT prior to the total remodel on 10-25-12.
New 3s: The current TT, with Vilemaw, new items, etc.

IMO: In my opinion

If there's anything I missed, please let me know.

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.