The concept of groupoid is one of the means by which the twentieth century reclaims the original domain of application of the group concept. The modern, rigorous concept of group is far too restrictive for the range of geometrical applications envisaged in the work of Lie. There have thus arisen the concepts of Lie pseudogroup, of differentiable and of Lie groupoid, and of principal bundle – as well as various related infinitesimal concepts such as Lie equation, graded Lie algebra and Lie algebroid – by which mathematics seeks to acquire a precise and rigorous language in which to study the symmetry phenomenae associated with geometrical transformations which are only locally defined.

This book is both an exposition of the basic theory of differentiable and Lie groupoids and their Lie algebroids, with an emphasis on connection theory, and an account of the author's work, not previously published, on the abstract theory of transitive Lie algebroids, their cohomology theory, and the integrability problem and its relationship to connection theory.

The concept of groupoid was introduced into differential geometry by Ehresmann in the 1950's, following his work on the concept of principal bundle. Indeed the concept of Lie groupoid – a differentiable groupoid with a local triviality condition – is, modulo some details, equivalent to that of principal bundle. Since the appearance of Kobayashi and Nomizu (1963), the concept of principal bundle has been recognized as a natural setting for the formulation and study of general geometric problems; both the theory of G-structures and the theory of general connections are set in the context of principal bundles, and so too is much work on gauge theory.