  Let $h$ be a three times partially differentiable function on $R^n$, let
$X=(X_1,\dots,X_n)$ be a collection of real-valued random variables and let
$Z=(Z_1,\dots,Z_n)$ be a multivariate Gaussian vector. In this article, we
develop Stein's method to give error bounds on the difference $E h(X) - E h(Z)$
in cases where the coordinates of $X$ are not necessarily independent, focusing
on the high dimensional case $n\to\infty$. In order to express the dependency
structure we use Stein couplings, which allows for a broad range of
applications, such as classic occupancy, local dependence, Curie-Weiss model
etc. We will also give applications to the Sherrington-Kirkpatrick model and
last passage percolation on thin rectangles.
