  In this article, we study the Brezis-Nirenberg type problem of nonlinear
Choquard equation with Neumann boundary condition \begin{equation*}
\begin{aligned} -\Delta u &= \lambda \alpha(x)u +
\left(\int\limits_{\Omega}\frac{u(y)^{2^*_{\mu}}}{|x-y|^{\mu}}\;dy\right)u^{2^*_{\mu}-1},
\;\;\text{in} \; \Omega,\\ \frac{\partial u}{\partial \nu} &= 0\;\; \text{on}
\; \partial\Omega, \end{aligned} \end{equation*} where $\Omega$ is a bounded
domain in $\mathbb{R}^N$ $(N\geq 4)$, $\nu$ is the unit outer normal to
$\partial \Omega$ and $\mu \in (0, N)$. According to the parameter $\lambda$,
we prove necessary and sufficient conditions for the existence and
non-existence of positive weak solutions to the problem. The proof is based on
variational arguments.
