This is my build on Anivia, I hope I can help everyone that likes to play Anivia become even better.

For Runes i go
9x Magic pen Marks
9x Mana Regen Seals
9x Cooldown Glyphs
3x Flat ability power Quints

This is a basic caster rune build with the Magic pen for extra damage, Mana regen for lane durability, Cooldown for well, Cooldown, and flat ability for more damage.

I go 9/0/21 getting magic pen in Offense and less cooldowns and more regen utilites.
This will allow you too stay in lane longer and farm more.

I go Maki Pendant first because Anivia can have some mana issues, after that i get my boots and Chalice, Boots for spell spen and Chalice for far greater mana regen, next up is Rabadons Deathcap for the majoir magic damage, then deathfire for the cooldown and damage as well. Thats as far as ive got with Anivia, please feedback for any tips on bettering the Items.

I level up FlashFrost as much as I can, to reduce the cooldown, Flash Frost is essentual for Anavia's combo and thus must have a short cooldown,
I then max Frostbite next, because of the chill effect that Anivia's abilites do, you will deal double damage, and is needed for nuking down enemies,
I don't get Crystalize till late game, I just feel it isn't as useful as it could be at ealier levels, therefore, I get it last,
as usual, I upgrade the ultimate Glacial Storm whenever possible.

For Summoner Spells I go Flash and Clarity,
Anivia doesn't have any staple escape abilities therefore flash is essentail for survival.
due to Anivia's mana issues, you'll need clarity to have enough mana for a kill or teamfight.

Anivia is really good in team fights, due to her Area Of Effect Ultimate, Glacial Storm, plus if she shoots Flash Frost and you explode it near a group of enemies, it will stun all of them, therefore making Anivia a great team Champion

As for farming, you'll want to focus on last hitting a lot, due to the gold cost of rabadons you'll want a lot of gold, therefore last hitting on Anivia is a good idea.

Note: I like to use her ultimate to farm minons, even though I run out of mana.

Pros:
Extreme Damage output at late game.
A good Solo and team Champion.
Very fun to play, even funnier when you wall someone out.
Can cheat Death with her passive, Rebirth, :)
People will call you ARTICUNO :D

Cons:
Extremely Squishy, low hp and low armour means if she gets focus, she'll die very quickly.
If you miss Flash Frost, you'll have to wait till it's cooldown before nuking someone properly.
The build will cost a bit, therefore, getting gold is essentail.
Even though her passive can revive her, it is not to be counted on, you will die very fast when Rebirth is activated, because it cannot withstand a multitude of attacks.

Anivia is a very fun champion to play as and can get you many kills if you play her right, her damage output is extremly high and you'll have fun seeing the enemies health drop. getting some survivabilty items may not be a bad idea,

Anivia and I would like to thank you for reading this build, it was a great pleasure typing it and having her cooling the room down on this hot summer day was a life saver :)

Anivia would like to say, that she is thankful for all the summoners that enjoy playing as her and would like to see herself being played as a lot more and to please leave some feedback.

Anivia: Others may be a light summer breeze for you, but with me, you'll be up against a howling wall of Ice.