  We use a new method via $p$-Wasserstein bounds to prove Cram\'er-type
moderate deviations in (multivariate) normal approximations. In the classical
setting that $W$ is a standardized sum of $n$ independent and identically
distributed (i.i.d.) random variables with sub-exponential tails, our method
recovers the optimal range of $0\leq x=o(n^{1/6})$ and the near optimal error
rate $O(1)(1+x)(\log n+x^2)/\sqrt{n}$ for $P(W>x)/(1-\Phi(x))\to 1$, where
$\Phi$ is the standard normal distribution function. Our method also works for
dependent random variables (vectors) and we give applications to the
combinatorial central limit theorem, Wiener chaos, homogeneous sums and local
dependence. The key step of our method is to show that the $p$-Wasserstein
distance between the distribution of the random variable (vector) of interest
and a normal distribution grows like $O(p^\alpha \Delta)$, $1\leq p\leq p_0$,
for some constants $\alpha, \Delta$ and $p_0$. In the above i.i.d. setting,
$\alpha=1, \Delta=1/\sqrt{n}, p_0=n^{1/3}$. For this purpose, we obtain general
$p$-Wasserstein bounds in (multivariate) normal approximations using Stein's
method.
