Abstract: When a group of linear transformations acts on a vector space, it is very natural to ask about properties of the quotient space. I'll explain David Hilbert's work on this problem, how it nearly killed a very popular area of 19th century mathematics, and why invariant theory is more relevant today than ever before. This talk reports on some of my research conducted while I was on sabbatical at the University of Maryland and the Research Institute of Symbolic Computation, Austria.