In this paper we prove the strict convexity, the interior C1,α,
C2,α and W2,p estimates for convex solutions to the Monge-Amp`ere
type equation. For the strict convexity and C1,α estimate, we assume
that the inhomogeneous term f satisfies a doubling condition. For the
C2,α and W2,p estimates, we assume that f is H¨older continuous or
continuous. These estimates are mainly due to Caffarelli. We also give a
brief discussion on the regularity for more general Monge-Amp`ere type
equations arising in optimal transportation.