The following papers constitute a study of the precise geometrical nature of the Cuspoid Catastrophes, the Elliptic, Hyperbolic and Parabolic Umbilics and the Reduced Double Cusp. Stereographic-pair figures permit a 3-dimensional realization of the Catastrophe Manifolds of the Cuspoid Catastrophes when observed through a stereo viewer. In introducing these precise pictures into the literature, it is hoped that they will aid in the application of Catastrophe Theory to the understanding of complex systems such as those of Biology, Physical Chemistry, Economics and Sociology.