This set of lectures has two primary objectives. The first one is to present the general theory of first order bifurcation that occur for vector fields in finite dimensional space. Illustrations are given of higher order bifurcations. The second objective, and probably the most important one, is to set up a framework for the discussion of similar problems in infinite dimensions.Parabolic systems and retarded functional differential equations are considered as illustrations and motivations for the general theory. Readers familiar with ordinary differential equations and basic elements of nonlinear functional analysis will find that the material is accessible and the fundamental results in bifurcation theory are presented in a way to be relevant to direct application. Most of the expository material consists of a concise presentation of basic results and problems in structural stability. The most significant contribution of the book is the formulation of structural stability and bifurcation in infinite dimensions. Much research should come from this - indeed some have already picked up the ideas in their work.