Abstract: In this expository talk I will describe some of the ideas used to establish localization for 1D lattice Schrödinger operators with random as well as quasiperiodic potentials. The approach is based upon the availability of large deviations type estimates for the iterates of the corresponding linear cocycle and the positivity of the maximal Lyapunov exponent. As a future goal we plan to investigate the viability of this approach for the mixed random-quasiperiodic model which Ao Cai, Pedro Duarte and I are currently studying.