  This paper is devoted to the investigation of the following nonlocal
dispersal equation $$
u_{t}(t,x)=\frac{D}{\sigma^m}\left[\int_{\Omega}J_\sigma(x-y)u(t,y)dy-u(t,x)\right]+f(t,x,u(t,x)),
\quad t>0,\quad x\in\overline{\Omega}, $$ where $\Omega\subset\mathbb{R}^{N}$
is a bounded and connected domain with smooth boundary, $m\in[0,2)$, $D>0$ is
the dispersal rate, $\sigma>0$ characterizes the dispersal range,
$J_{\sigma}=\frac{1}{\sigma^{N}} J\left(\frac{\cdot}{\sigma}\right)$ is the
scaled dispersal kernel, and $f$ is a time-periodic nonlinear function of
generalized KPP type. We first study the principal spectral theory of the
linear operator associated to the linearization of the equation at $u\equiv0$.
We obtain an easily verifiable and general condition for the existence of the
principal eigenvalue as well as important sup-inf characterizations for the
principal eigenvalue. We next study the influence of the principal eigenvalue
on the global dynamics and confirm the criticality of the principal eigenvalue
being zero. It is then followed by the study of the effects of the dispersal
rate $D$ and the dispersal range characterized by $\sigma$ on the principal
eigenvalue and the positive time-periodic solution, and prove various
asymptotic behaviors of the principal eigenvalue and the positive time-periodic
solution when $D,\sigma\to0^{+}$ or $\infty$. Finally, we establish the maximum
principle for time-periodic nonlocal operator.
