Hi, my name is Captain Inšanø and i have been playing league since season 4. anivia was the first champ i learned and i have been playing her ever since. over several accounts i probably have over 2000 games played with her.
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Anivia is a bursty control mage. she excels in group fight and is weak when fighting alone. anivia has a decent laning phase but should focus on farming so she can get as much gold (and in turn items) before team fighting happens. Anivia is a good champ regardless of if your team is ahead or behind. she is great at tower hugging or seiging.
I find anivias damage has dropped off this season but with reworks to her cooldowns and kit i find she is better at controlling the battle field. she has low cool downs on her ult and with proper stuns and walls you can control exactly where you enemey walks or stop them entirely.

Early game anivia is very weak in my opinion. her mana pool is not big enough to support fighting and her damage is not all that great. her cc is average at best as she only has the one stun and her small wall.
early on you want to focus hard on getting your manaflow stacks maxed so you are constantly getting mana back. i try to aim for full stack and level 6 before my first back. once you have full stacks you can clear a wave quick with your ult, back, and get back to lane while missing little to no cs.
i try to get vision on both sides of my river so i can ping my team if my laner is heading their way. if laners do decide to roam then you can shove your lane hard with your ult and get tower plates to punish them. this will often force the other team back to their mid lane. I usually run tp to come in to assist lanes if they need it. i find roaming as anivia is risky as she is quite squishy and getting jumed on usually results in her death and loss of her egg passive (its very weak early on)

Rely on anivias passive to stay in lane longer. if you dont have enough gold yet you can bait the enemy into killing you under your tower. they wotn be able to keep attacking you, and you can spawn back with full health.

Early game with anivia is all about getting as much gold as possible so when team fights come around you can shine.
ANIVIA IS ALL ABOUT TEAM FIGHTS!
just farm as much as possible and shove for turret plates when you can

Mid game is when you should start looking to join in on fights.
by this time you have the mana and the damage to start helping your team. you can shove waves hard with your ult and look to gank other lanes, or just stick to farming to further better yourself for late game.
at this point in the game though you should be helping with objectives and getting control of more of the map. anivia is great for helping hold tower as her ult can clear a whole minion wave before they even get to the tower. she can also seige well by placing her ult in the enemies safe zone under their tower. this will either push the enemy back (allowing you to hit thier tower risk free), or force them forward into you, and then you can hit them without entering turret range.

Late game anivia is a beast. at this point you should always be with your team. even though you are amazing in team fights. solo, you can still easily be killed.
when in groups, your team can peel them off your egg when you do get killed.
often the enemy will focus you, as you do insane AOE damage and can split their team with walls while stunning them. having your team around to peel you will help hugely and ensure you can deal the damage your character was designed to do

fighting with anivia is all about dealing as much damage as possible while holding the enemy off of you and your back line as possible.
when they enemy engages you can Ult right in the middle of their team to seperate them. if they do stay grouped then you can wall half their team off and force them to fight.
using the blizzard and walls make the enemy extremely easy to hit with your stun as well.
proper use of your blizzard in fights can turn the whole battle. either the enemy keeps fighting you (and loses lots of life and movement speed) ro they run away and allow you to get free hits. a poorly placed blizzard prevents any of this from happening.
it takes time but learning to use all your abilities appropriately are what serperates great from bad anivias. know how to use your wall and stuns to keep the enemy in your blizzard just a couple seconds longer is the difference between getting 1 and 3 E's in during a fight. it can be the difference between fighting their whole team or just 2 of them at a time.

during baron and drakes i tend to just auto attack the objecting.
i will place my ult at a choke point to prevent the enemies from coming and engaging use. if they do decide to walk through the blizzard, stuns are ver ez to hit and follow up with an E. at this point the enemy will be below half life and ez for your team to jump on.
if several of their team come through the blizzard you can wall them off and pick off the ones that enter.
if the enemy team decides to come in through a different choke point you can always wall them off for 4-5 secdonds, cancel your ult and swithc it to the choke point theyre entering

PLEASE LET ME KNOW IF YOU HAVE ANY QUESTIONS OR WANT MORE INFO OF ANY SORT ON ANIVIA
I HAVE LOTS OF PLAY TIME ON HER AND CAN HELP WITH MOST QUESTIONS