We prove existence of a positive radial solution to the Choquard equation $$-Δu +V u=(I_α\ast |u|^p)|u|^{p-2}u\qquad\text{in}\,\,\,Ω$$ with Neumann or Dirichlet boundary conditions, when $Ω$ is an annulus, or an exterior domain of the form $\mathbb{R}^N\setminus \bar{B}_a(0)$. We provide also a nonexistence result, that is if $p\ge\frac{N+α}{N-2}$ the corresponding Dirichlet problem does not have any nontrivial regular solution in strictly strictly star-shaped domains.