A ribbon is a double structure on P-1. The geometry of a ribbon is closely related to that of a smooth curve. In this paper we consider linear series on ribbons. Our main result is an explicit determinantal description for the locus W-2n(r) of degree 2n line bundles with at least (r + 1)-dimensional sections on a ribbon. We also discuss some results of Clifford and Brill-Noether type.