  For smooth projective curves the Clifford index is an important invariant
which provides a bound for the dimension of the space of sections of a line
bundle. This is the first step in distinguishing curves of the same genus. In
this paper we generalise this to introduce Clifford indices for semistable
vector bundles on curves. We study these invariants, giving some basic
properties and carrying out some computations for small ranks and for general
and some special curves. For curves whose classical Clifford index is two, we
compute all values of our new Clifford indices.
