A differential module is a module equipped with a square-zero endomorphism. By developing a theory for deforming an arbitrary free complex into a differential module, we classify (up to quasi-isomorphism) the free differential modules whose homology is isomorphic to a given module M. We use an iterative approach to parameterize the deformations and obstructions in terms of certain Ext groups, and apply this theory to study certain rigidity properties of free resolutions and related rank conjectures.