  This paper studies the nonlinear stochastic partial differential equation of
fractional orders both in space and time variables: \[
\left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) =
I_t^\gamma\left[\rho(u(t,x))\dot{W}(t,x)\right],\quad t>0,\: x\in\mathbb{R}^d,
\] where $\dot{W}$ is the space-time white noise, $\alpha\in(0,2]$,
$\beta\in(0,2)$, $\gamma\ge 0$ and $\nu>0$. Fundamental solutions and their
properties, in particular the nonnegativity, are derived. The existence and
uniqueness of solution together with the moment bounds of the solution are
obtained under Dalang's condition:
$d<2\alpha+\frac{\alpha}{\beta}\min(2\gamma-1,0)$. In some cases, the initial
data can be measures. When $\beta\in (0,1]$, we prove the sample path
regularity of the solution.
