  The dimension of spaces of global sections for line bundles on semistable
curves parametrized by the compactified Picard scheme is studied. The theorem
of Riemann is shown to hold. The theorem of Clifford is shown to hold in the
following cases: the curve has two components; the curve is any semistable
curve and the degree is either 0 or 2g-2; the curve is stable, free from
separating nodes, and the degree is at most 4. These results are all shown to
be sharp. Applications to the Clifford index, to the combinatorial description
of hyperelliptic curves, and to plane quintics are given.
