  We study the Wasserstein distance $W_2$ for Gaussian samples. We establish
the exact rate of convergence $\sqrt{\log\log n/n}$ of the expected value of
the $W_2$ distance between the empirical and true $c.d.f.$'s for the normal
distribution. We also show that the rate of weak convergence is unexpectedly
$1/\sqrt{n}$ in the case of two correlated Gaussian samples.
