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We consider the deformation of a discontinuous group acting on the Euclideanspace by affine transformations.  A distinguished feature here is that even a `small' deformation of a discretesubgroup may destroy proper discontinuity of its action. In order to understandthe local structure of the deformation space of discontinuous groups, weintroduce the concepts from a group theoretic perspective, and focus on`stability' and `local rigidity' of discontinuous groups.  As a test case, we give an explicit description of the deformation space ofZ^k acting properly discontinuously on R^{k+1} by affine nilpotenttransformations.  Our method uses an idea of `continuous analogue' and relies on the criterionof proper actions on nilmanifolds

DEFORMATION OF PROPERLY DISCONTINUOUS ACTIONS OF ℤk ON ℝk+1

DEFORMATION OF PROPERLY DISCONTINUOUS ACTIONS OF ℤ^k ON ℝ^k+1