  Riemann surface carries a natural line bundle, the determinant bundle. The
space of sections of this line bundle (or its multiples) constitutes a natural
non-abelian generalization of the spaces of theta functions on the Jacobian.
There has been much progress in the last few years towards a better
understanding of these spaces, including a rigorous proof of the celebrated
Verlinde formula which gives their dimension. This survey paper tries to
explain what is now known and what remains open.
