This talk is about joint work with Ted Chinburg and Luc Illusie on a generalization to complexes of Mazur's deformation theory for modules for a profinite group. A new question arises when deforming complexes: Is the versal deformation represented by a bounded complex of modules which are each finitely generated over the versal deformation ring? We will describe some evidence that this is the case when the complex arises from arithmetic. This talk is sponsored by a grant from the Fund to Encourage Women (FEW) of the Trustees' Council of Penn Women.