Twentyfirst Arnold Sommerfeld Lecture Series, 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.