In this lecture, I will present results on how to measure quantitatively the quality of a coarse-graining procedure in computational statistical physics. We typically have in mind two types of coarse-graining techniques: starting from dynamics in a high dimensional space (molecular dynamics with a large number of atoms), one would like either to project the dynamics onto a few collective variables, or to approximate the dynamics by a jump Markov models between the metastable conformations. The former approach is related to the construction of effective dynamics using the Mori-Zwanzig formalism, and to the computation of free energy using free energy adaptive biasing procedures (metadynamics, adaptive biasing force). The latter approach is useful to understand the foundation of Markov State models (or kinetic Monte Carlo), and of the accelerated molecular dynamics à la D. Perez and A.F Voter. From a mathematical viewpoint, the former requires functional inequalities (Poincaré or logarithmic Sobolev inequalities) while the latter relies on the notion of quasi-stationary distribution.