The study of ordinary and symbolic powers of ideals is related to the study of blowing up of varieties, singularities in positive characteristic, integral closures and normalization of rings, among others. It is desirable to retrieve homological information about these powers. We consider homological invariants related to free resolutions of powers of ideals, for example the depth and Castelnuovo-Mumford regularity. An important goal is to describe the possible values for the sequence of depth (or regularity) of such powers.