This central chapter revolves around Mather measuresmeasureMather. We give two equivalent definitions of those measures, prove their existence and relate their support with the Aubry sets. We then use those measures to prove the convergence of solutions to the discounted equations.discounted equation This convergence result is also generalized to degenerate discounted equations. Finally we study similarities and differences of discrete and continuous Mather measures in the classical Lagrangian setting. The same is done for the discrete and the continuous discounted equations.