A crossed product of a ring and a group is essentially a semidirect product of the two, arising from an action of the group as automorphisms of the ring. Important examples occur in geometry when a group acts on a space, and thus on any algebra of functions on the space. In that setting, there are connections between the cohomology and deformation theories of the space and of the crossed product. In this talk we will discuss some examples of crossed products and their Hochschild cohomology, and give some general results on the ring structure of Hochschild cohomology of crossed products. We will also describe an example of a deformation of a crossed product arising from a universal deformation formula based on a finite quantum group, and speculate on the proper general setting for this example.