In this talk, we first give a brief introduction to the classical Aubry Mather theory and the Mather theory for higher dimensional Hamiltonian systems. Then we report on some recent results for quasi-periodic Lagrangian systems on the annulus. This is the middle ground between classical twist maps and higher dimensional systems. In particular, we prove that for any rotation number r, the set of action minimizing orbits of this rotation class is not empty, then we prove the existence of a diffusive orbit when there is no flow-invariant curves on the annulus. This is a generalization of Mather's connecting orbit on the annulus for twist maps.