I wont say this is the ultimate katarina build , but i use this very often and works well.
If you think you know better items for katarina then please put them in the comment section.

Katarina needs alot of magic penetration so she can kill her enemy's fast with her ultimate.
She needs some HP aswell so she can survive when she tower dives with shunpo.

I use Ignite and Ghost to get more AP and movement speed to dive towers

Alot of people will have a comment about this.
I use this 10/0/20 so i get magic penetration , cooldowns and the ignite cooldown buff.

These items will give you enough survive ability and alot of magic damage.
Dont buy the Soulstealer if you cant kill a enemy often.

-Katarina's Passive is amazing , it resets your cooldowns with 15 seconds when you make a assist or a kill. you can tower dive an low hp champion at a tower with shunpo and when you kill him , you can use shunpo on a minion or champion that stands away from the tower to get away fast.
-The thing you will try to do is hitting your opponent between his skill cooldowns to evade unneccesary damage.
-Katarina's bouncing blade will bounce to enemys who are near.
-On low level your dagger doesnt bounce to alot of enemy,s so try to last hit minions with your dagger.
-You can harras an enemy champion when he is near 1 of his minions.
-Try to keep your enemy around 50% of his health so the risk he will try to kill you will be small.
-If your enemy has a crowd control like the following : stun , fear or silence you have to perfectly time your Death Lotus between the cooldowns.

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