Tamagawa numbers of algebraic groups and geometry

We will begin by recalling the definition of the Tamagawa number of an algebraic group. In the late 1970's, Harder posed a conjecture relating this invariant to moduli spaces of G-bundles. The general form of this conjecture remains open. Harder's conjecture was reinterpreted using the trace formula for stacks by Behrend. This approach settled the conjecture in certain special cases. We will end with a discussion of recent work of Heinloth and Schmitt.