Let K be an arbitrary field of characteristic 0, and the n-dimensional affine space over K. A well-known cancellation problem asks, given two algebraic varieties with isomorphic cylinders and , whether and themselves are isomorphic. In this paper, we focus on a related problem: given two varieties with equivalent (under an automorphism of ) cylinders and , are and equivalent under an automorphism of ? We call this stable equivalence problem. We show that the answer is positive for any two curves . For an arbitrary , we consider a special, arguably the most important, case of both problems, where one of the varieties is a hyperplane. We show that a positive solution of the stable equivalence problem in this case implies a positive solution of the cancellation problem.