  Let $X$ be an irreducible smooth projective curve of genus $g\ge3$ defined
over the complex numbers and let ${\mathcal M}_\xi$ denote the moduli space of
stable vector bundles on $X$ of rank $n$ and determinant $\xi$, where $\xi$ is
a fixed line bundle of degree $d$. If $n$ and $d$ have a common divisor, there
is no universal vector bundle on $X\times {\mathcal M}_\xi$. We prove that
there is a projective bundle on $X\times {\mathcal M}_\xi$ with the property
that its restriction to $X\times\{E\}$ is isomorphic to $P(E)$ for all
$E\in\mathcal{M}_\xi$ and that this bundle (called the projective Poincar\'e
bundle) is stable with respect to any polarization; moreover its restriction to
$\{x\}\times\mathcal{M}_\xi$ is also stable for any $x\in X$. We prove also
stability results for bundles induced from the projective Poincar\'e bundle by
homomorphisms $\text{PGL}(n)\to H$ for any reductive $H$. We show further that
there is a projective Picard bundle on a certain open subset $\mathcal{M}'$ of
$\mathcal{M}_\xi$ for any $d>n(g-1)$ and that this bundle is also stable. We
obtain new results on the stability of the Picard bundle even when $n$ and $d$
are coprime.
