This self-contained monograph presents a new approach to scattering theory for hyperbolic differential equations. Applications to potential scattering and the Schrodinger equation are included. One of the main results is the meromorphic character of the scattering matrix. The mathematical tools used are a combination of spectral theory, harmonic analysis, theory of semigroups, and various aspects of the theory of partial differential equations. This monograph will be of interest to mathematicians and mathematical physicists. It is also suitable for use as a text in advanced courses or seminars.