As an effect of confinement, quark masses cannot be measured directly in experiments. Nonetheless their values can be indirectly defined in a rigorous way. In this talk, after discussing extensively various possible definitions of the quark masses, I will concentrate on those suitable for fully non-perturbative analysis of the theory, and review the results obtained by modern lattice approaches, highlighting in particular the recent progress made on the computation of the up-down quark mass difference and on the heavy quark masses.