  In this article we present an $L_p$-theory ($p\geq 2$) for the
time-fractional quasi-linear stochastic partial differential equations (SPDEs)
of type $$ \partial^{\alpha}_tu=L(\omega,t,x)u+f(u)+\partial^{\beta}_t
\sum_{k=1}^{\infty}\int^t_0 ( \Lambda^k(\omega,t,x)u+g^k(u))dw^k_t, $$ where
$\alpha\in (0,2)$, $\beta <\alpha+\frac{1}{2}$, and $\partial^{\alpha}_t$ and
$\partial^{\beta}_t$ denote the Caputo derivative of order $\alpha$ and $\beta$
respectively. The processes $w^k_t$, $k\in \mathbb{N}=\{1,2,\cdots\}$, are
independent one-dimensional Wiener processes defined on a probability space
$\Omega$, $L$ is a second order operator of either divergence or non-divergence
type, and $\Lambda^k$ are linear operators of order up to two. The coefficients
of the equations depend on $\omega (\in \Omega), t,x$ and are allowed to be
discontinuous. This class of SPDEs can be used to describe random effects on
transport of particles in medium with thermal memory or particles subject to
sticking and trapping.
