  In this paper, we first consider two scalar nonlocal diffusion problems with
a free boundary and a fixed boundary. We obtain the global existence,
uniqueness and longtime behaviour of solution of these two problems. The
spreading-vanishing dichotomy and sharp criteria for spreading and vanishing
are established. We also prove that accelerated spreading could happen if and
only if a threshold condition is violated by kernel function. Then we discuss a
classical Lotka-Volterra predator-prey model with nonlocal diffusions and a
free boundary which can be seen as nonlocal diffusion counterpart of the model
in the work of Wang (2014 J. Differential Equations \textbf{256}, 3365-3394).
