Classical Brill-Noether theory studies, for given g, r, d, the space of line bundles of degree d with r+1 global sections on a curve of genus g. We will review the main results of this theory, and then we will discuss the situation for higher-rank vector bundles, where even the most basic questions remain wide open. We will discuss the role of spaces with fixed determinant, and if time permits, how the dimension theory of Artin stacks can improve on current existence results.