Singularity theory is not a theory in the usual (axiomatic) sense. Indeed it is precisely its width, its vague boundaries and its interaction with other branches of mathematics, and science, which makes it so attractive. Our subject then rather defies a neat definition. This nebulous nature can, however, lead to identity crises and I then find it useful to think of singularity theory as the direct descendant of the differential calculus. It has, for example, the same concerns with Taylor series, and one can view much current research as natural extensions of problems which our forefathers laboured with, and considered central. The calculus is the tool, par excellence (sadly my only concession to the French language) for studying physics, differential equations in general and the geometry of curves and surfaces. Consequently one might hope that singularity theory will have applications in these fields. Indeed one might judge the vigour of the subject by its success in these areas.
In this paper I will focus on applications to the differential geometry of surfaces in Euclidean and projective 3-space, for three reasons. First its familiarity: this is a classical area known to all professional mathematicians, with a high intuitive content. Secondly any originality here is a good measure of success, this is a well worn path to take. And finally because it illustrates in a quite striking way the benefits which accrue when one moves from studying linear and quadratic phenomena to those of higher order.