Lie groupoids and Lie algebroids.- Connections on Lie groupoids and Lie algebroids.- Groupoids of fibre morphisms.- Four case studies.- Symmetries.- Cartan geometries.- A comparison with alternative approaches.- Infinitesimal Cartan geometries on TM.- Projective geometry: the full version.- Conformal geometry: the full version.- Developments and geodesics.- Cartan theory of second-order differential equations.