  The first half of this paper is concerned with the topology of the space
$\AAA(M)$ of (not necessarily contact) Anosov vector fields on the unit tangent
bundle $M$ of closed oriented hyperbolic surfaces $\Sigma$. We show that there
are countably infinite connected components of $\AAA(M)$, each of which is not
simply connected. In the second part, we study contact Anosov flows. We show in
particular that the time changes of contact Anosov flows form a $C^1$-open
subset of the space of the Anosov flows which leave a particular $C^\infty$
volume form invariant, if the ambiant manifold is a rational homology sphere.
