In the first part of this paper a rigorous proof of the application of the method of stationary phase to double and multiple integrals is established with the aid of neutralizer or unitary functions. It is shown that the principal contributions to U(k) come from small but otherwise arbitrary neighbourhoods of critical points of the integral, which may be located in the interior or on the boundary of the domain of integration. These points are associated with the phase or amplitude function. An explicit asymptotic series in the parameter k of the principal contribution is exhibited when the amplitude and the phase functions have in the neighbourhood of a critical point (x1,y1) a development of the form g(x,y) = (x−x1)λ0−1 (y−y1)μ0−1g1(x,y), π(x,y) = π(x1,y1) + aσ,0 (x−x1σ [1 + P(x,y) + b0,τ(y−y1τ[1 +Q(x,y)]. The function g1 is a regular function and P,Q can be developed in power series in the vicinity of the critical point and vanish at this point. The above expansion we shall call normal or canonical and the critical point a normal or canonical critical point of the integral. Although the assumption of the normal form expansion of the amplitude and phase functions is too restrictive for the general case, nevertheless it is found to be sufficiently broad to include most of the important and interesting cases which occur in diffraction, scattering and other problems of mathematical physics. In Part II the principal contribution arising from a critical point of normal type has been calculated in the form of a descending power series in the parameter k. It is shown, with the use of majorant functions, that the contribution due to the remainder part of the series is of higher order in the parameter than that of the last term of the finite part, which proves the asymptotic character of the series in the sense of Poincaré. The results derived here are in agreement with that of Part I. However, the new series has a decided advantage over that given in Part I if calculations are desired for even a few terms of the series, since the coefficients entering in the asymptotic expansion of the principal contribution are expressed directly in terms of the original functions g(x,y) and π(x,y) and their derivatives, which is not the case in the formulas derived in Part I. In Part III explicit asymptotic expansions of the double integral are derived for several typical critical points associated with the phase function. These are important in connection with the theory of diffraction of optical instruments with large aberrations and scattering problems. On account of their importance, each case has been treated in detail. In the appendices we have given an alternative proof of the theorem announced in Part I and the derivation of the leading term due to a boundary stationary point. There will be found also a discussion of the more general integral where the parameter k appears implicitly in the phase function and not explicitly as considered in the text. Integrals of this kind occur in many branches of physics, especially when dealing with wave propagation in dispersive and absorbing media. Finally, we have concluded on the basis of our results that the Rubinowicz approach to diffraction and the stationary phase application to diffraction integrals lead to similar mathematical results, although different physical interpretations, in diffraction phenomena, the former leading to Young diffraction phenomena and the latter to Fresnel diffraction phenomena.