A very extensive review of quantum algorithms, and more importantly, an exploration on what type of mathematical problems that can be more efficiently solved using computer computers versus classical computers.
Abstract: Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation and, in particular, on problems with an algebraic flavor.
The arXiv version can be found here.
 A.M. Childs and W. van Dam, Rev. Mod. Phys. v.82, p.1 (2010).