  These are the notes of a short course I gave in the school "Aspects
m\'etriques et dynamiques en analyse complete", Lille, May 2015. The aim of
this notes is to describe how to use a geometric structure (namely, the
Kobayashi distance) to explore and encode analytic properties of holomorphic
functions and maps defined on complex manifolds. We shall first describe the
main properties of the Kobayashi distance, and then we shall present
applications to holomorphic dynamics in taut manifolds, strongly pseudo convex
domains and convex domains, and to operator theory in Bergman spaces (Carleson
measures and Toeplitz operators).
