  In this paper, we study the following coupled Choquard system in $\mathbb
R^N$: $$\left\{\begin{align}&-\Delta u+A(x)u=\frac{2p}{p+q} \bigl(I_\alpha\ast
|v|^q\bigr)|u|^{p-2}u,\\ &-\Delta
v+B(x)v=\frac{2q}{p+q}\bigl(I_\alpha\ast|u|^p\bigr)|v|^{q-2}v,\\ &\ u(x)\to0\ \
\hbox{and}\ \ v(x)\to0\ \ \hbox{as}\ |x|\to\infty,\end{align}\right.$$ where
$\alpha\in(0,N)$ and $\frac{N+\alpha}{N}<p,\ q<2_*^\alpha$, in which
$2_*^\alpha$ denotes $\frac{N+\alpha}{N-2}$ if $N\geq 3$ and $2_*^\alpha :=
\infty$ if $N=1,\ 2$. The function $I_\alpha$ is a Riesz potential. By using
Nehari manifold method, we obtain the existence of positive ground state
solution in the case of bounded potential and periodic potential respectively.
In particular, the nonlinear term includes the well-studied case $p=q$ and
$u(x)=v(x)$, and the less-studied case $p\neq q$ and $u(x)\neq v(x)$. Moreover
it seems to be the first existence result for the case of $p\neq q$.
