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.