Linear series on semistable curves

For a semistable curve X of genus g, the number h^0(X, L) is studied for line bundles L of degree d parameterized by the compactified Picard scheme. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: X has two components; X is any semistable curve, and d=0 or d=2g − 2; X is stable, free from separating nodes, and d at most 4. These results are shown to be sharp. Applications to the Clifford index, to the combinatorial description of hyperelliptic curves, and to plane quintics are given.