A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^γX(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $γ\in (0,\infty)$, 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 $A := L^β$ and $Q:=\tilde{L}^{-α}$ 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-)Matérn fields to space-time.