Local states break translational invariance and previous descriptions considered in the book have to be adapted. The local states in general give origin to scattering of the modes of the translational invariant system and may lead to bound states. Also, these bound states may be intrinsic to some impurities added to a given system. Significant example are the Anderson model and the Kondo model, here discussed. Due to the presence of internal states of the impurities, their presence induces many-body effects with significant consequences. When the number of impurities increases, one may tend to a system that is translational invariant but in general multiband, such as the Anderson lattice. The presence of local states leads in general to strong local correlations between electrons and perturbative methods have to be replaced by non-perturbative methods to handle the strong correlations.