This is a joint work with G. Courtois, S. Gallot and A. Sambusetti. We will study some compact metric spaces satisfying weak versions of curvature assumptions. More precisely we will use two invariants : entropy (growth of the measure of balls in the universal cover) and the diameter to describe a compactness result for a class of compact metric spaces. The class consists in spaces satisfying curvature assumptions that can be defined only with triangles comparison such as CAT(0)-spaces or $\delta$-hyperbolic spaces in the sense of Gromov. A key ingredient is a variant of a celebrated inequality relating volumes of balls of different radii centred at the same point and called the Bishop-Gromov inequality. The talk is intended to be elementary, for non experts and all notions should be defined.