  We consider an evolution equation involving the fractional powers, of order
$s \in (0,1)$, of a symmetric and uniformly elliptic second order operator and
Caputo fractional time derivative of order $\gamma \in (1,2]$. Since it has
been shown useful for the design of numerical techniques for related problems,
we also consider a quasi--stationary elliptic problem that comes from the
realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map
for a nonuniformly elliptic problem posed on a semi--infinite cylinder. We
provide existence and uniqueness results together with energy estimates for
both problems. In addition, we derive regularity estimates both in time and
space; the time--regularity results show that the usual assumptions made in the
numerical analysis literature are problematic
