Hyperbolic boundaries vs. hyperbolic groups

The aim of these notes is to connect the theory of hyperbolic and relatively hyperbolic groups to the theory of manifolds and Kleinian groups. We also give definitions and many examples of relatively hyperbolic groups and their boundaries. We survey some of the extensive work that has been done in the field. These notes are based on lectures given by the third author at CIRM in the Summer of 2018.