  The basic problem of optimal transportation consists in minimizing the
expected costs $\mathbb {E}[c(X_1,X_2)]$ by varying the joint distribution
$(X_1,X_2)$ where the marginal distributions of the random variables $X_1$ and
$X_2$ are fixed. Inspired by recent applications in mathematical finance and
connections with the peacock problem, we study this problem under the
additional condition that $(X_i)_{i=1,2}$ is a martingale, that is, $\mathbb
{E}[X_2|X_1]=X_1$. We establish a variational principle for this problem which
enables us to determine optimal martingale transport plans for specific cost
functions. In particular, we identify a martingale coupling that resembles the
classic monotone quantile coupling in several respects. In analogy with the
celebrated theorem of Brenier, the following behavior can be observed: If the
initial distribution is continuous, then this "monotone martingale" is
supported by the graphs of two functions $T_1,T_2:\mathbb {R}\to \mathbb {R}$.
