The analysis of electromagnetic scattering from a large, deep, and arbitrarily-shaped open cavity is a very difficult task because of a large number of unknowns resulting from the discretization of the cavity. In this paper, we present a very efficient technique to deal with this problem. This technique is based on the finite-element method that is known for its capability to handle arbitrary geometries and complex material composition. The technique exploits the unique features of the finite-element equations and, more importantly, the unique features of the problem of scattering by a large and deep cavity. It is designed in such a manner that it uses a minimal memory, which is proportional to the aperture size of the cavity and independent of the depth of the cavity, and its computation time increases only linearly with the depth of the cavity. Furthermore, it computes the scattered fields for all angles of incidence without requiring significant additional time. The technique is applied to both two and three dimensional cavities whose interior may be coated with radar absorbing materials. Excellent results are obtained for two dimensional cavities and those for three dimensional cavities are also promising.