Clifford's theorem and higher rank vector bundles

We give here a refinement of the classical Clifford's theorem for the upper bound of the number of independent global sections of a semistable vector bundle on a smooth curve. We also conjecture a new version of this theorem that takes into account the Clifford index of the curve. In the case of a bi-elliptic curve we obtain a very precise bound. Finally we study the case of rank 2 bundles.