This talk will present an overview of recent progress towards a solution of one
of the grand-challenges of modern science: understanding the properties of
interacting electrons in molecules and solids. After an introduction to the
physics I will argue our theoretical understanding of a basic model system,
the two dimensional Hubbard model, has reached the level that we can say
with confidence that its superconducting properties capture key aspect of the
high-Tc superconductivity in copper-oxide materials. I will then summarize
the current status of our extension of the methods to fully physically realistic
systems, emphasizing the areas of theoretical uncertainty and the prospects
for resolution.