Inspired by brane constructions of quiver gauge theories, a year and a half ago Davide Gaiotto proposed a vast generalization: families of 4D N=2 superconformal theories parametrized by the moduli space of genus-g Riemann surfaces with punctures. The construction involves compactifying the 6D (2,0) SCFT on a curve, C, with certain defect operators at the punctures. I will describe a procedure for classifying the resulting superconformal theories . Any curve, C, on which the 6D A_{N-1} SCFT is compactified, can be decomposed into 3-punctured spheres, connected by cylinders. We will classify the spheres, and the cylinders that connect them. The classification is carried out explicitly, up through N=5, and for several families of SCFTs for arbitrary N. These lead to a wealth of new S-dualities between Lagrangian and non-Lagrangian 4D N=2 SCFTs.