Like any other major advance, catastrophe theory depends on a number of concepts with which the reader new to the field is unlikely to be familiar. The aim of this chapter is to provide an introduction to these which is as simple as possible. A more sophisticated, but still relatively accessible, account is given by Poston & Stewart (1978a). Mathematicians might prefer Lu (1976) or, for full proofs, Brocker & Lander (1975) or Trotman & Zeeman (1976).