  A new class of fractional-order stochastic evolution equations of the form
$(\partial_t + A)^\gamma X(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $\gamma \in
(0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a
separable Hilbert space $H$ and the spatiotemporal driving noise $\dot{W}^Q$ is
the formal time derivative of an $H$-valued cylindrical $Q$-Wiener process.
Mild and weak solutions are defined; these concepts are shown to be equivalent
and to lead to well-posed problems. Temporal and spatial regularity of the
solution process $X$ are investigated, the former being measured by mean-square
or pathwise smoothness and the latter by using domains of fractional powers of
$A$. In addition, the covariance of $X$ and its long-time behavior are
analyzed.
  These abstract results are applied to the cases when $A := L^\beta$ and
$Q:=\tilde{L}^{-\alpha}$ are fractional powers of symmetric, strongly elliptic
second-order differential operators defined on (i) bounded Euclidean domains or
(ii) smooth, compact surfaces. In these cases, the Gaussian solution processes
can be seen as generalizations of merely spatial (Whittle-)Mat\'ern fields to
space-time.
