First the method of stationary phase is used to develop various semiuniform asymptotic techniques that can be used to calculate the field in regions in which diffraction catastrophes occur. Three types of catastrophes are treated, namely those that take place at a shadow boundary, at a caustic and at a cusp. Then the diffraction of a spherical wave by a circular aperture is considered, and it is shown how the use of standard diffraction theories leads to diffraction integrals that can be evaluated by means of the semi-uniform asymptotic techniques developed for the three diffraction catastrophes mentioned above. Finally, it is pointed out how the asymptotic results provide new physical insight into the diffraction process, as well as offering great savings in computing time.