For more than two centuries, integral geometry, also called the theory of geometric probabilities, has followed the development of probability, measure theory and geometry.

The first chapter recalls historical results: Buffon's needle and Bertand's paradoxes.

Chapters 2 to 7 and 10 present the basic notions and methods: slice using all affine lines or planes, project on all affine lines and planes, and average. The method is applied to curves in \({\mathbb R}^2\) or \({\mathbb R}^3\) and surfaces in \({\mathbb R}^3\). In fact results in dimensions 2 and 3 extend easily to higher dimensions and to space-forms. The last section of Chapters 7, 8, and 9 are devoted to statements of the form “topology implies some lower bound on the total curvature”. Again the objects are curves or surfaces contained in \({\mathbb R}^3\).

Integral geometry in spheres, Lorentz space of dimension 3 or hyperbolic space \({\mathbb H}^3\) are the topics of Chapters 11, 13, 15 and 16. Chapter 12 deals with integral geometry of foliations of a domain of \({\mathbb R}^3\) or of \({\mathbb S}^3\).

The compacity of Grassmann manifolds was essential to obtain formulas of Euclidean integral geometry in \({\mathbb R}^3\) or of \({\mathbb S}^3\). The situation changes drastically when dealing with extrinsic conformal geometry of curves or foliations in \({\mathbb S}^3\) or \({\mathbb R}^3\). This is the topic of Chapters 17 and 18.

Chapter 19 is an isolated point of the book. Its goal is to understand the geometry of the levels of a complex polynomial of two variables near an isolated singular point. An appendix briefly presents some notions used in the book.

Pictures are an essential part of the book and often contain the main ideas of the proofs.