This book addresses the problem of artificial boundaries, ie boundaries which have no physical counterpart. It describes the question of absorbing boundary conditions in fluid mechanics, and includes many interesting related contributions in wave propagation. There are two situations where artificial boundaries are introduced. First, for computing in a bounded region by a problem formulated on an infinite domain and secondly, in the context of domain decomposition methods. Several methods were developed including: integral equations, infinite elements, boundary layers, and fictitious boundaries with adapted boundary conditions. The latter have become very popular in geophysics and aeronautics.