  This is largely a survey paper, dealing with Cartan geometries in the complex
analytic category. We first remind some standard facts going back to the
seminal works of F. Klein, E. Cartan and C. Ehresmann. Then we present the
concept of a branched holomorphic Cartan geometry which was introduced by the
authors in [BD]. It generalizes to higher dimension the notion of a branched
(flat) complex projective structure on a Riemann surface introduced by
Mandelbaum. This new framework is much more flexible than that of the usual
holomorphic Cartan geometries (e.g. all compact complex projective manifolds
admit branched holomorphic projective structures). At the same time, this new
definition is rigid enough to enable us to classify branched holomorphic Cartan
geometries on compact simply connected Calabi-Yau manifolds.
