Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structure of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of ``unknown shift'' problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure., Comment: 12 pages, 2 figures, requires amsmath, amssymb, graphicx, psfrag and theorem packages. To appear in the proceedings of the ACM-SIAM Symposium on Discrete Algorithms 2003. This article supersedes the earlier reports quant-ph/0011067 and quant-ph/0205034