  In this paper we show that Musielak--Orlicz spaces are UMD spaces under the
so-called $\Delta_2$ condition on the generalized Young function and its
complemented function. We also prove that if the measure space is divisible,
then a Musielak--Orlicz space has the UMD property if and only if it is
reflexive. As a consequence we show that reflexive variable Lebesgue spaces
$L^{p(\cdot)}$ are UMD spaces.
