The most basic notion of Catastrophe Theory is introduced in this chapter, namely, that of a critical point of a smooth function. This notion is also referred to in the standard literature as a singular point or a singularity. An essential distinction is made by dividing these points into two classes—the non-degenerate and the degenerate critical points. The degenerate ones are more difficult to handle and will be discussed in later chapters, whereas the nondegenerate critical points can be completely treated here.