This survey describes a class of methods known as "fast direct solvers". These algorithms address the problem of solving a system of linear equations Ax=b arising from the discretization of either an elliptic PDE or of an associated integral equation. The matrix A will be sparse when the PDE is discretized directly, and dense when an integral equation formulation is used. In either case, industry practice for large scale problems has for decades been to use iterative solvers such as multigrid, G...