This book was originally intended to be largely disjoint from my earlier book Lie groupoids and Lie algebroids in differential geometry, [Mackenzie, 1987a], written in the period from 1980 to mid 1985 and published in the London Mathematical Society Lecture Note series in 1987. However I have included, in Part II of the present book, the central chapters of the earlier book on transitive Lie algebroids, their cohomology and connection theory and their integrability. Despite the dramatic results since 2000 on the integrability problem for general Lie algebroids, and work on more sophisticated cohomology theories, it seems to me well worth while to continue to treat the case of locally trivial Lie groupoids and transitive Lie algebroids independently of the general theory. As I have noted already, the systematic use of Lie groupoids and abstract Lie algebroids provides a thoroughgoing reformulation of standard connection theory, and is likely to retain its own character independent of the more general results. This material has in some cases been rewritten and in others left almost unchanged, though typos and obscurities have, I hope, always been caught.

The earlier book was intended to be readable without a detailed prior knowledge of connection theory, and certainly without any acquaintance with groupoids or Lie algebroids, and was consequently leisurely in pace. I feel it is no longer necessary to argue throughout the book for the importance of groupoids in differential geometry, and I have now also assumed that readers have a basic knowledge of connection theory and principal bundles, as well as the standard processes of manifolds, vector bundles and Lie groups.