The integral cohomology ring of a toric manifold has an elegant description in terms of the underlying fan or characteristic function. Orbifolds and singular toric varieties on the other hand, can be somewhat intractable. Even so, when certain checkable criteria are satisfied, their integral cohomology can often have a characterization as pleasing as in the smooth case. This outcome results whenever the orbifold can be constructed by attaching appropriate “cells” in a manner analogous to a CW complex. This is a report of joint work with Dietrich Notbohm, Soumen Sarkar and Jongbaek Song.