Notes for Lectures at the Fabes-Riviere Symposium

Abstract. These notes discuss results on layer potential methods for elliptic boundary problems, with emphasis on the Dirichlet problem for the Laplace opera-tor. They start by reviewing results for domains with moderately smooth boundary, then for Lipschitz domains, and proceed to discuss results in [HMT], obtained with S. Hofmann and M. Mitrea, for a class of domains we call regular Semmes-Kenig-Toro (SKT) domains, often called chord-arc domains with vanishing constant, and for ε-regular SKT domains, often called chord-arc domains with small constant. Contents 1