This is an introductory text dealing with a part of mathematics: modern differential geometry and the theory of Lie groups. It is written from the perspective of and mainly for the needs of physicists. The orientation on physics makes itself felt in the choice of material, in the way it is presented (e.g. with no use of a definition–theorem–proof scheme), as well as in the content of exercises (often they are closely related to physics).

Its potential readership does not, however, consist of physicists alone. Since the book is about mathematics, and since physics has served for a fairly long time as a rich source of inspiration for mathematics, it might be useful for the mathematical community as well. More generally, it is suitable for anybody who has some (rather modest) preliminary background knowledge (to be specified in a while) and who desires to become familiar in a comprehensible way with this interesting, important and living subject, which penetrates increasingly into various branches of modern theoretical physics, “pure” mathematics itself, as well as into its numerous applications.

So, what is the minimal background knowledge necessary for a meaningful study of this book? As mentioned above, the demands are fairly modest. Indeed, the required mathematical background knowledge does not go beyond what should be familiar from standard introductory undergraduate mathematics courses taken by physics or even engineering majors. This, in particular, includes some calculus as well as linear algebra (the reader should be familiar with things like partial derivatives, several variables Taylor expansion, multiple Riemann integral, linear maps versus matrices, bases and subspaces of a linear space and so on).