ABSTRACT: Machinery is developed to establish the existence and stability of double-pulse solutions of reaction-diffusion systems of partial differential systems on the line, given the existence of a stable single-pulse solution. A double-pulse solution consists of two separated pulses, each closely approximating the single pulse. The method is from dynamical systems theory; the system is reduced to a system of ordinary differential equations, where pulse solutions correspond to homoclinic solutions, which are investigated geometrically. Existence is established by consideration of certain return (Poincaré) maps. Stability is intimately associated with an orientation, which in turn is associated with a direction of crossing of certain stable and unstable manifolds associated to the homoclinic solutions. General machinery is developed and applied to the case at hand, and an illustrative example from population dynamics is presented.