In this book we first review the ideas of Lie groupoid and Lie algebroid and the associated
concepts of connection. We next consider Lie groupoids of fibre morphisms of a fibre bundle
and the connections on such groupoids together with their symmetries. We also see how the
infinitesimal approach using Lie algebroids rather than Lie groupoids and in particular using
Lie algebroids of vector fields along the projection of the fibre bundle may be of benefit.We
then introduce Cartan geometries together with a number of tools we shall use to study them.
We take as particular examples the four classical types of geometry: affine projective
Riemannian and conformal geometry. We also see how our approach can start to fit into a more
general theory. Finally we specialize to the geometries (affine and projective) associated
with path spaces and geodesics and consider their symmetries and other properties.
