in the systems move randomly between the adjacent spatial locations (see [@BrCoFl; @CaCo; @CaCo1; @CaCo2; @Co; @CoCuPo; @Fisher; @Ha; @He; @He0; @HeKo; @HessWein; @Kolmo; @Zhaox], etc.). Nonlocal dispersal operators, such as $u(t, \cdot)\mapsto  \int_{{\ensuremath{\mathbb{R}}}^N}k(y-\cdot)[u(t, y)-u(t, \cdot)]dy$, are applied in modeling diffusive systems in applied sciences when the systems exhibit long range internal interactions (see [@FeIsPe; @Fif; @GrHiHuMiVi; @HuGr; @HuMaMiVi; @KaLoSh1; @OtDuAl; @ShZh0; @St], etc.). Diffusive evolution equations with both random and nonlocal dispersals have been widely studied on one hand. On the other hand, there are still many important dynamical issues for such systems which are not well understood yet.

The current paper is devoted to the study of principal spectrum points/principal eigenvalues of nonlocal dispersal operators with time periodic indefinite weight functions and Dirichlet type, Neumann type, and periodic boundary conditions. More precisely, the eigenvalue problem subject to Dirichlet type boundary condition considered in this paper reads as $$\label{dirichlet-ori}
\begin{cases}
 -\p_t u(t, x)+\int_D \kappa(y-x)u(t, y)dy-u(t, x)+\lambda_1 m_1(t,x)u(t, x)=0,\quad x\in \bar D,\\
u(t+T, x)=u(t, x),
\end{cases}$$ where $D\subset \R^N$ is a smooth bounded domain and $m_1(t, x)\neq 0$ is a continuous, time periodic weight function; the eigenvalue problem subject to Neumann type boundary condition considered in this paper reads as 