Amumu vs Master Yi

Save a charge of Bandage Toss (Q) after your engage: the spell has two, and the second catches an enemy who escapes with a dash or Flash.

Wait until several enemies are in range to cast Curse of the Sad Mummy (R): it stuns every enemy around you and curses them.

Keep Despair (W) on against tanks: every second, it deals a percentage of their max health to nearby enemies.

Full clear with Despair (W) and Tantrum (E), then gank: both hit every nearby monster, and Tantrum's cooldown drops when you get hit.

Against Highlander (R), don't run alone: regroup with your team, because he gains speed and ignores slows.

If you have CC other than a slow, land it right after Alpha Strike (Q): during the strike, he's untargetable, and Highlander (R) ignores slows.

If you have CC, interrupt Meditate (W): he channels it to heal, more at low health, and to reduce the damage he takes.

Put pressure on his jungle early on: Master Yi is strongest once he has his items, and each stolen camp delays him.