Over the last years propagating beam methods have gained increasing acceptance for solving a broad range of wave propagation problems. The transformation of an extremely difficult boundary value problem into a much easier initial value problem, has allowed the solution of problems that were considered as intractable. The development of those propagating beam methods has occurred simultaneously, but almost independently, in the fields of acoustics and electromagnetics (primarily integrated and fiber optics). This paper reviews and assesses the main advantages and problems associated with the use of this method. Applications to acoustics and optics, to linear and nonlinear materials are considered and illustrated by examples.