  We present novel martingale concentration inequalities for martingale
differences with finite Orlicz-$\psi_\alpha$ norms. Such martingale differences
with weak exponential-type tails scatters in many statistical applications and
can be heavier than sub-exponential distributions. In the case of one
dimension, we prove in general that for a sequence of scalar-valued
supermartingale difference, the tail bound depends solely on the sum of squared
Orlicz-$\psi_\alpha$ norms instead of the maximal Orlicz-$\psi_\alpha$ norm,
generalizing the results of Lesigne & Voln\'y (2001) and Fan et al. (2012). In
the multidimensional case, using a dimension reduction lemma proposed by
Kallenberg & Sztencel (1991) we show that essentially the same concentration
tail bound holds for vector-valued martingale difference sequences.
