The purpose of this book is to give a quick introduction to the theory of foliations, as well as to Lie groupoids and their infinitesimal version – Lie algebroids. The book is written for students who are familiar with the basic concepts of differential geometry, and all the results presented in this book are proved in detail.

The topics in this book have been chosen so as to emphasize the relations between foliations, Lie groupoids and Lie algebroids. Lie groupoids form the main tool for the study of the ‘transversal structure’ (the space of leaves) of a foliation, by means of its holonomy groupoid. Foliations are also a special kind of Lie algebroids. At the same time, the elementary theory of foliations is a very useful tool in studying Lie groupoids and Lie algebroids.

In Chapter 1 we present the basic definitions, examples and constructions of foliations. Chapter 2 introduces the notion of holonomy, which plays a central role in this book. The Reeb stability theorems are discussed, as well as Riemannian foliations and their holonomy. This chapter also contains an introduction to the theory of orbifolds (or V-manifolds). Orbifolds provide a language to describe the richer structure of the space of leaves of certain foliations; e.g. the space of leaves of a Riemannian foliation is often an orbifold.

In Chapter 3 we present two classical milestones of the theory of foliations in codimension 1, namely the theorems of Haefliger and Novikov, with detailed proofs. Although the proofs make essential use of the notion of holonomy, this chapter is somewhat independent of the rest of the book (see the figure).