Heise
Block building with almost nothing to memorise: squares, edges, then a commutator for the corners.
Ryan Heise's method solves the cube by understanding rather than algorithms: square blocks built where they are, joined and edge-oriented, then every edge and all but three corners, which a commutator finishes. It is efficient in moves, which made it a model for fewest-moves solving, and slow to execute at speed, because every step is planned.
Squares
Build square blocks (a corner with two edges and the centres between them), attaching each where it fits best. The squares need not match in colour yet, so you can use whatever blocks the scramble already half-built.
Matching and edge orientation
Match the squares up, if they are not matched already, and orient the remaining edges. Beginners do these separately; with practice they become one step.
Edges and two corners
Solve every remaining edge and any two of the remaining corners, again two steps at first and one later.
Last three corners
The last corners, wherever they are, are solved with a commutator (see Commutators and conjugates). When they all sit in the last layer, the case is an L3C case.
Heise: the whole walkthrough
Squares
Matching and edge orientation
Edges and two corners
Last three corners
All systems