Classical bifurcation theory is undergoing revitalization with the infusion of ideas from singularities of mappings and structural stability. The situation now is similar to that a quarter century ago when Krasnosel'skii introduced topological methods, especially degree theory, into the subject. Like degree theory, the theory of singularities of mappings is playing a fundamental role in the development of the subject.
Our goal is to give a few examples of how qualitative ideas can give insight into bifurcation problems. The literature and full scope of the theory is too vast to even attempt to survey here.
