Hello, and welcome to my Anivia guide.

My name is M4nk3y and I play on the European Server. Anivia has been my main ever since I reached level 30 and I still play her rather often. I have over 500 games with her, and reached 1800 Elo while playing her. The first build is a build for Summoner's Rift and the second build is a build for Twisted Treeline (I don't play this often).
Anivia is a very strong champion that I count as a Tier 1 AP Carry in both Summoner's Rift as in Twisted Treeline.

My runes are obvious choices in Summoner's Rift.

- Magic Penetration Marks are locked on Anivia.

- Flat Mana Regen is good for early game, and the scaling Mana Regen is just to fill in for later in the match.

- Cooldown Reduction Glyphs grant us 34.89% Cooldown Reduction with Frozen Heart, which is very close to the cap. If you obtain the Golem Buff (Which you should almost always have) you'll be at the 40.00% cap anyways.

- Flat Ability Power Quintessences are good for Early Game lane domination.

For Twisted Treeline I recommend tankier runes since Anivia is already ridiculously strong from items and her nature.

Standard Caster Masteries that I would use on every Caster. Awareness over Expanded Mind because Expanded Mind is a horrible Mastery (5% increased Mana Pool which is counted over your BASE maximum mana).

The items are quite mana-based. If you look at them, 3 of them grant Mana, and actually alot, which makes Archangel's Staff alot stronger. I would recommend Boots of Speed for early game if you're laning against someone with skillshots, but you can also just get a Mana Crystal and rush Catalyst for an amazing laning phase.

In Twisted Treeline I get the Moonflair Spellblade to get some Tenacity. If your teammates are good enough they won't let you die easily and you can hide in the back while dealing sick amounts of damage.

In Summoner's Rift I focus damage first with Flash Frost right after Frostbite. Frostbite is Anivia's ability I maximize first, since this deals a good amount of damage when someone is frozen and it's not a skillshot. A good tip is to place your ultimate under someone and then Frostbiting him, and quickly removing your ultimate, dealing a great amount of damage without to much bother. Aiming your Flash Frost is way harder, and that's why I max this second. Standard order:

R>E>Q>W

Full order: Q-E-E-W-E-R-E-Q-E-Q-R-Q-Q-W-W-R-W-W

In Twisted Treeline I take a level of Crystallize earlier (level 8) so I can block off paths in the jungle easier, which is very important for Dragon Control.

Full order: Q-E-E-W-E-R-E-W-E-Q-R-Q-Q-Q-W-R-W-W

Flash is a must.

Ignite can be swapped with what I recommend: Ghost and Teleport. Don't forget to spec your mastery in the Summoner Spell you will use.
I personally don't think any other Summoner Spell is useful.

Thank you for reading my guide, vote as your wish accordingly, and leave a comment (If you want to vote AND comment, do the commenting part in the vote, since it's required to comment to vote till 20 votes are done).

Hints and tips:
1. Ask for the Golem Buff on Summoner's Rift if your jungler does not need it anymore (Most of the times they will give it you after their first Golem)
2. You can use your Crystallize to place people in your Glacial Storm or lead them into your Flash Frost. This may require a lot of practice.
3. Putting a few points into Crystallize earlier will make the wall larger and in this way you can close of roads easier, especially in the jungle.
4. When you're on the run and both you and your enemy are low on HP, you can quickly turn around and take down your enemy, since you'll have enough burst for this most of the times.

That's all I can think off right now, good luck playing Anivia, she's a really strong Champion.

Cheers,

M4nk3y