Over the past two decades, the geometry of surfaces and, more generally, submanifolds in Möbius geometry has (re)gained popularity. It was probably T. Willmore's 1965 conjecture [306] that stimulated this increased interest: Many geometers have worked on this conjecture, and in the course of this work it turned out (see for example [67] and [40]) that the Willmore conjecture is in fact a problem for surfaces in Möbius geometry and that the corresponding local theory was already developed by the classical geometers (cf., [218]). A crucial classical reference was [29]; however, it may not be very easy to obtain and, once found, may not be very easy to read, especially for non-German—speaking colleagues.

A similar story could be told about the recent developments on isothermic surfaces — here, it was the relation with the theory of integrable systems, first pointed out in [71], that made the topic popular again; also in this case [29] turned out to be a treasure trove, but many more results are scattered in the classical literature.

The present book has a twofold purpose:

1. — It aims to provide the reader with a solid background in the Möbius geometry of surfaces and, more generally, submanifolds.

2. — It tries to introduce the reader to the fantastically rich world of classical (Möbius) differential geometry.