  We give a Cram\'{e}r moderate deviation expansion for martingales with
differences having finite conditional moments of order $2+\rho, \rho \in
(0,1],$ and finite one-sided conditional exponential moments. The upper bound
of the range of validity and the remainder of our expansion are both optimal.
Consequently, it leads to a "half-side" moderate deviation principle for
martingales. It is worth mentioning that our result is new even for independent
random variables. Moreover, applications to quantile coupling inequality,
$\beta$-mixing and $\psi$-mixing sequences are discussed.
