Although a radiation or diffraction problem may have a unique solution, certain formulations of the problem may result in integral equations that do not have solutions, or whose solutions are not unique. Typically, when considering the exterior problem, trouble may occur at frequencies corresponding to resonance in the interior of the body. This paper summarizes our present state of knowledge concerning existence and uniqueness. The implications in connection with approximate numerical solution of the integral equations are examined. Methods are discussed for obtaining integral equations that always have unique solutions. The theoretical advantages and disadvantages of various methods will be compared.