A (proper) cubulation of a group G is a CAT(0) cube complex
X together with a proper action of G on X. Partial cubulation drops
the properness assumption. Sometimes a partial cubulation can be
promoted to a proper cubulation. Sometimes a proper cubulation can be
improved by passing through a partial cubulation. I'll give examples
of theorems of both types, focusing on hyperbolic and relatively
hyperbolic groups. This is joint work with Daniel Groves.