  The regularity of the free boundary in optimal transportation is equivalent
to that of the potential function along the free boundary. By establishing new
geometric estimates of the free boundary and studying the second boundary value
problem of the Monge-Amp\`ere equation, we obtain the $C^{2,\alpha}$ regularity
of the potential function as well as that of the free boundary, thereby resolve
an open problem raised by Caffarelli and McCann in \cite{CM}.
