  We develop an $\e$-regularity theory at the boundary for a general class of
Monge-Amp\`ere type equations arising in optimal transportation. As a corollary
we deduce that optimal transport maps between H\"older densities supported on
$C^2$ uniformly convex domains are $C^{1,\alpha}$ up to the boundary, provided
that the cost function is a sufficient small perturbation of the quadratic cost
$-x\cdot y$.
