Chaos Theory is a synonym for dynamical systems theory, a branch of mathematics. Dynamical systems come in three flavours: flows (continuous dynamical systems), cascades (discrete, reversible, dynamical systems), and semi-cascades (discrete, irreversible, dynamical systems). Flows and semi-cascades are the classical systems introduced by Poincare a century ago, and are the subject of the extensively illustrated book: Dynamics: The Geometry of Behaviour, Addison-Wesley 1992 authored by Ralph Abraham and Shaw. Semi- cascades, also know as iterated function systems, are a recent innovation, and have been well-studied only in one dimension (the simplest case) since about 1950.The two-dimensional case is the current frontier of research. And from the computer graphics of the leading researcher come astonishing views of the new landscape, such as the Julia and Mandelbrot sets in the beautiful books by Heinz-Otto Peigen and his co-workers. Now, the new theory of critical curves developed by Mira and his students and Toulouse provide a unique opportunity to explain the basic concepts of the theory of chaos and bifurcations for discrete dynamical systems in two-dimensions.The materials in the book and on the accompanying disc are not solely developed only with the researcher and professional in mind, but also with consideration for the student. The book is replete with some 100 computer graphics to illustrate the material, and the CD-ROM contains full-colour animations that are tied directly into the subject matter of the book, itself. In addition, much of this material has also been class-tested by the authors. The cross-platform CD also contains a software program called ENDO, which enables users to create their own 2-D imagery with X-Windows. Maple scripts are provided which give the reader the option of working directly with the code from which the graphics in the book were given.