In this chapter, we deal with dynamical systems to which we apply the foregoing results. In particular, we give a spectral characterization of the differentmixing properties (weak, mild, strong). All the results are well-known, and we omit the classical proofs for which we refer to [61,93,111,139,145,151,193,197,200,241] or others. We close this chapter by an overview on group extensions over an ergodic rotation.