This is a series of lectures on Bishop-Gromov's type inequalities adapted to metric spaces. We consider the case of Gromov-hyperbolic spaces and draw consequences of these inequalities such as compactness and finiteness Theorems. This course is intended to be elementary in the sense that the necessary background is described in detail.The final version of this article will appear in the book « Surveys in Geometry » (ed. A. Papadopoulos), Springer Nature, 2021.