The object of this book is to introduce to a new generation of students an area of mathematics that has received a tremendous impetus during the last twenty years or so from developments in singularity theory.

The differential geometry of curves, families of curves and surfaces in Euclidean space has fascinated mathematicians and users of mathematics since Newton's time. A minor revolution in mathematical thought and technique occurred during the 1960s, largely through the inventive genius of the French mathematician René Thorn. His ideas (partly inspired by the earlier researches of H. Whitney) gave birth to what is now called singularity theory, a term which includes catastrophes and bifurcations. Not only has singularity theory made precise sense of what many of the earlier writers on differential geometry were groping to say (as so often happens, their instinct was uncannily good but they lacked the proper formal setting for their ideas); it has also made possible a richness of detail that would have stirred the imagination of any of the great geometers of the past.

Thom applied his ideas to many fields besides geometry, for example in his famous (but very difficult) book (Thorn, 1975).