The cohomology ring of the moduli space of stable holomorphic vector bundles of rank $n$ and degree $d$ over a Riemann surface of genus $g > 1$ has a standard set of generators when $n$ and $d$ are coprime. When $n = 2$ the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when $n > 2$ and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.