The study of singular points of algebraic curves in the complex plane is a meeting point for many different areas of mathematics. The beginnings of the study go back to Newton. During the nineteenth and early twentieth century algebraic geometers working on plane curves developed methods which allowed them to deal with singular curves: see e.g. [168], [84], and several articles in volume III of the mathematical encyclopaedia published 1906–1914. A notable achievement was the resolution of singularities of such curves. In the late 1920s results in the then new area of topology were applied to the knots and links in the 3-sphere obtained by looking at the neighbourhood of such a singularity. There was a resurgence of interest about 1970 due to the interaction with newly developing ideas from singularity theory in higher dimensions, most importantly, the fibration theorem which Milnor had just discovered, in the context of functions of several complex variables. There has been continuous development since then, a particular point of interest being the application of Thurston's (circa 1980) decomposition theorems for 3-manifolds and for homeomorphisms of 2-manifolds.

The interaction between ideas from these different sources makes the study of curve singularities particularly fruitful and exciting. Equisingularity is an equivalence relation which admits characterisations from numerous differing points of view. The development of the ideas leading up to this is the leitmotif of the first half of this book. I thus emphasise the equivalence of different approaches, and feel that many results gain in clarity from appearing in an integrated account.