Impurity models and the development of quantum cluster approaches to lattice models of quantum condensed matter

My aim in this talk will be to provide a historical perspective on the evolution of computational quantum many-body ideas and methods, starting from the first numerical solutions of quantum impurity problems leading on to the development of modern quantum cluster theories of strongly correlated and disordered systems, which is the subject of this symposium. I will review some of the advances that took place in the solution of quantum impurity models in the 70s and 80s, the development of the dynamical mean-field theory in the late 80s and early 90s, and of its cluster extensions in the late 90s and the early 2000s, both in terms of the techniques that were involved and the understanding they led to.