  Let ${\cal S}{\cal U}(r, L_0)$ denote the moduli space of semi stable vector
bundles of rank $r$ and fixed determinant $L_0$ of degree $d$ on a smooth curve
$C$ of genus $g \geq 3$. In this paper we describe the group of automorphisms
of $ {\cal S}{\cal U}(r, L_0) $. The analogue of this result is carried out for
the space ${\cal U}(r,d) $ of semi stable vector bundles of rark $r$ and degree
$d$. As an application of the technics we use, we give in the appendix at the
end of the paper a proof of the Torelli theorem for the moduli spaces ${\cal
S}{\cal U}(r, L_0) $ for any rank $r$ and degree $d$.
