  In this paper, we prove that the tangent bundle of the moduli space
$\cSU_C(r,d)$ of stable bundles of rank $r>2$ and of fixed determinant of
degree $d$ (such that $(r,d)=1$), on a smooth projective curve $C$ is always
stable, in the sense of Mumford-Takemoto. This verifies a well-known
conjecture, and is related to a conjectural existence of a K\"ahler-Einstein
metric on Fano varieties with Picard number one.
