In many applications in wave propagation and scattering one has to solve the wave equation in the frequency domain. Problems are deemed ``high-frequency'' if the domain contains many wavelengths. The linear systems which arise from finite element approximation are sparse, complex, non-Hermitian and generally considered to be ``highly indefinite''. The iterative solution of these systems in optimal time is an outstanding unsolved problem in numerical linear algebra. We present an overview of recent progress on the development and analysis of domain decomposition preconditioners for these problems, where the preconditioner is constructed from the corresponding problem with added absorption. Our preconditioners incorporate local subproblems that can have various boundary conditions, and include the possibility of a global coarse mesh. We give rigorous field of values estimates which lead to GMRES convergence theory for problems with absorption. The construction of good solvers for the ``pure'' Helmholtz case without absorption is done by combining the theoretical insight obtained with numerical experimentation. The theory is for problems with constant wavespeed. Some emerging new results on the variable wavespeed case are mentioned at the end of the talk.