How the Horadric Cube works
The Horadric Cube is found in Act 2 (Halls of the Dead, Dry Hills) and doubles as a portable stash and a crafting bench: drop the ingredients inside, hit Transmute, and the listed result comes out. In D2R every recipe works in all game modes — the old ladder-only restrictions from classic Diablo II are gone, so single-player and non-ladder characters can use everything on this page.
Not all 121 recipes are equally useful. The ones players actually rely on: rune upgrades for cleaning up mid runes, the four socketing recipes (and the Hel unsocket), unique and set item upgrades (exceptional and elite), the Grand Charm reroll (any magic item plus three perfect gems), repair and recharge recipes for charged runewords, and the Cow Level and uber portals. Gem upgrades matter mostly to feed the other recipes — perfect gems are crafting fuel.
Recipes worth memorizing
If you only learn a handful: three of the same rune (two from Um up) makes the next rune; three same-tier gems make the next tier; a normal unsocketed weapon or armor plus two runes and a perfect gem rolls sockets; Hel plus a Scroll of Town Portal clears sockets at the cost of whatever is inside; any magic item plus three perfect gems rerolls it — the recipe behind Grand Charm hunting; and Ral or Ort plus a damaged item repairs it without a trip to town.