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Playing as Karthus

Tips to maximize your impact

Aim for isolated champions
Aim Lay Waste (Q) at a champion away from their minions: if the blast hits only one enemy, it deals much more damage.
Wall before your combo
Place Wall of Pain (W) in their path before your combo: anyone who crosses it is slowed and loses magic resist.
Keep playing after you die
After you die, keep casting your spells: Death Defied (passive) keeps you in spirit form for a while, and you can still cast Lay Waste (Q) and Requiem (R).
Requiem for the finish
Save Requiem (R) to finish off wounded enemies, often right after a teamfight: it hits every enemy champion wherever they are, after a 3-second channel.

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Global strengths and weaknesses

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How to counter and exploit weaknesses

Stay near your minions
Stay near your minions or your allies: Lay Waste (Q) deals much more damage when the blast hits only one enemy.
Jump on him if you can
If you can engage, jump on him: Karthus has no dash or leap to escape.
Don't stay low once he hits 6
Once he hits level 6, back instead of staying at low health: Requiem (R) hits every enemy champion wherever they are, 3 seconds after he starts channeling.
Save your protection for Requiem
If you have a shield, a heal or invulnerability, save it for Requiem (R): it hits every enemy champion 3 seconds after he starts channeling.
Move away after he dies
Once Karthus is dead, move away: Death Defied (passive) keeps him in spirit form for a while, and he can still cast his spells.