  Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences.
Set $X_n=\sum_{i=1}^n \xi_i $ and $ \langle X \rangle_n=\sum_{i=1}^n
\mathbf{E}(\xi_i^2|\mathcal{F}_{i-1}).$ We prove Cram\'er's moderate deviation
expansions for $\displaystyle \mathbf{P}(X_n/\sqrt{\langle X\rangle_n} \geq x)$
and $\displaystyle \mathbf{P}(X_n/\sqrt{ \mathbf{E}X_n^2} \geq x)$ as
$n\to\infty.$ Our results extend the classical Cram\'{e}r result to the cases
of normalized martingales $X_n/\sqrt{\langle X\rangle_n}$ and standardized
martingales $X_n/\sqrt{ \mathbf{E}X_n^2}$, with martingale differences
satisfying the conditional Bernstein condition. Applications to elephant random
walks and autoregressive processes are also discussed.
