The aim of this paper is to show that a general inverse scattering formulation illuminates alternative, computationally efficient solution methods for several classes of signal processing problems. Inverse scattering problems arise in physics, transmission-line synthesis, geophysics and acoustics and in one class of formulations they require a procedure to determine the parameters of a layered wave propagation medium from measurements taken at the boundary. There exists a close relationship between the physical inverse scattering problems and some important issues in signal processing such as the design of digital filters, the development of linear prediction algorithms and their lattice filter implementations and cascade synthesis of systems with a given impulse response (realization problems). For many of these problems several efficient algorithms already exist in the literature, but the connection between the different solutions was not always clear. Recently, the push to VLSI implementations led to the realization that, in spite of their apparent similarity, the alternative algorithms possess radically different properties when, say, a parallel implementation is sought. In this paper we shall show that alternative procedures that are usually arrived at by various clever tricks, in fact correpond to two conceptually extremely simple, basic ways of solving inverse scattering problems: the so called "layer-peeling" and "layer-adjoining" methods. Examples include the Schur vs Levinson methods for determining the optimal filters for prediction of stationary stochastic processes, and the generalized Lanczos vs Berlekamp-Massey methods for the partial realization (Pade approximation) problem, and also several recent design procedures for some classes of digital filters.