In this paper, we obtain some regularities of the free boundary in optimal transportation with the quadratic cost. Our first result is about the $C^{1,alpha}$ regularity of the free boundary for optimal partial transport between convex domains for densities $f, g$ bounded from below and above. When $f, g in C^alpha$, and $partialOmega, partialOmega^*in C^{1,1}$ are far apart, by adopting our recent results on boundary regularity of Monge-Amp`ere equations cite{CLW1}, our second result shows that the free boundaries are $C^{2,alpha}$. As an application, in the last we also obtain these regularities of the free boundary in an optimal transport problem with two separate targets.
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