The talk will describe recently developed fast solvers for the linear systems arising upon the discretization of elliptic PDEs. While most existing fast methods tend to be based on iterative solvers such as GMRES, the new techniques directly construct an approximate inverse (or LU factorization) of the coefficient matrix. This makes the techniques robust and particularly fast for problems involving multiple right hand sides. Such "fast direct solvers" have been developed both for the sparse (and often very large) matrices that arise upon finite element discretizations of elliptic PDEs, and for the dense matrices arising upon discretization of the associated integral equations.