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Analytic families of differential complexes, which are useful for proving blowing down and direct image theorems for regular families of strongly pseudoconvex manifolds, are defined and shown to have an associated deformation theory. The results of this deformation theory include a criteria for the coherence of the cohomology sheaves of an analytic family of complexes, a theorem on the upper semi-continuity of the dimensions of the cohomology of the fibers of the family and a result on the constancy of the Euler characteristic along the fibers.