Time-harmonic waves are observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. As the frequency increases, the numerical solution of such problems becomes increasingly difficult, due to the large number of degrees of freedom required to resolve each oscillation. The memory requirements and asymptotic complexity of direct solvers are too costly for these problems, so iterative solvers must be used. However, most iterative methods have their own numerical difficulties; for boundary element methods, the dense matrix-vector multiplication needed at each iteration is very costly, while for finite element methods, the indefiniteness and ill-conditioning of the matrix destroy convergence. In this talk, I will present fast algorithms and preconditioners which allow the convergence of Krylov subspace iterative solvers for high-frequency problems in a reasonable amount of time.