Hi, Disp here with an updated Anivia build! Only having a 44% rating makes me sad and I can see why people dont like the old build, and I dont either. So thats why I remade this build, and you may see many like it. This build is from my first main and through the many months i played as her, i finally perfected my Anivia build.

Let me explain my build:
-I get a Meki Pendant first. This, for one, increases mana regen, which allows for more harassment, and two, it builds into the next item (Tear of Goddess), which I immediately get the next time I return to base. I also get an hp pot to reduce the chance of resorting to my passive. (Note: do NOT rely on this passive! Though it may save you sometimes, mid and late game it will pretty much be useless unless you have other to scare them off or a nearby turrets to hit those pesky squishies!) The less you use the passive, the better the chance it will be used in a somewhat safe situation.
-After I immedietaly go for the soulstealer since if you are good as anivia, then you will have those "i wish i had the SS right now!!" moments. However, if you find yourself constantly ganked or getting to your tower a little to late, get boots that will fit the game you are in (alot of CC, go mercury's treads or for added damage (dmg as i type it), go sorcerors shoes). Feel free to just get the boots of speed if you dont need that much speed just yet.
-get boots or if you already got those, get soulstealer if you dont have it.
-get Rod of Ages. Main reason: more hp for survivability and mp for longer fights. Fully building it will give you increasing hp, mp, and ap for as long as the game goes (with a cap, of course).
-after the RoA, i finish my Archangel's Staff for the added AP and mana/mana regen.
-if not gg there, go ahead and build a Rabadon's Deathcap for the major AP bonus.
-now if the game is still going on (one of those epic gg, awesome for both sides games), screw the enemy over with a void staff or build up whatever is needed for that game.

If you are really good (like i was), you should have 20 stacks or close to it, max RoA, and basically having the enemy raging. The main reason im so good is because i try to stay out of the major battle area while sniping them with Flash Frost and only go in when death is minimal, nuking everything in my way and resulting in some SS stacks and killsteals (your team will get over it if you are really good). Ignite, even though it may seem awesome, save it for 1v1's or 1v2's (where at least one kill is available and you have your passive ready with others on their way). Add it to your ultimate for maximum DPS. If you want, use it on a champ thats running from your allies. it'll help them, but i recommend saving it for when you can get the kill. Also, i am highly against getting aura items as it will require you to get close to your enemy for it to work and you have enough ap to make up for it.

Runes... um, yeah, get those runes. If you prefer to use different runes, go ahead, but i advise AP quints as they help the boost on the deathcap. Masteries are a standard 9/0/21 for mages, focusing on exp. gain more than more mana.

The skill sequence is set up for early game, the flash frost will be your major dmg dealer, but after you get the ultimate, the sequence is set for massing the dmg output for frostbite, as it will be your main dmg dealer for mid and late game. Also, for summoner spells, i get flash for a quick escape or to 'port in closer for that kill, and ignite for the reasons above. i max crystalize last since its not an offensive skill but i get it at lvl 4 to help run away from gankers. also, get used to aiming the wall since it has a slight delay with casting, as it will prove to be a handy offensive tool in many different situations. In battle, use your ult first to get them slowed for the double damage of Frostbite, and finish up with stunning the enemy(s) with your Flashfrost while they are still in your ult, and wall them in if needed.

Okay, um, thats it for information. Just remember to play smart; only initiate if you are able to survive and get a kill, stay away from major battle with -1/2 hp (with or without egg, its stupid to risk it) unless getting kills without dieing is a possibility, and never solo late-mid and late game (lvl 10+) as Anivia can be a very popular target to gank, especially if you're leading the game.

Unit Profile:
Anivia is what I call a "pure mage," meaning that she can easily get kills in a team fight or a 1v1 without the need for a gank or entering the team fight at the end to "mop up" any straglers. Also, she is on the slow side, even with boots, so it is higly recommended to stay with others who you know will protect you in case of a gank after the laning phase.

NOTE: This build, including the explaination, went through a major overhaul, so if there are any odd pieces of information that doesnt belong, ignore it. I dont need to know about small mistakes unless it makes understanding it more difficult.