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Abstract: Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes in terms of a so-called colon-operation. From this, generators of can be obtained in finite time if it is finitely generated. Under the additional hypothesis that is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is . Along the way, we develop some techniques for dealing with nonfinitely generated algebras. In particular, we introduce the finite generation ideal. [Copyright &y& Elsevier]