The potential functions in the optimal transportation satisfy a Monge-Ampere type equation. When the cost function c(x, y)=|x-y|^2, it is the standard Monge-Ampere equation, and has been studied by many people. For more general cost functions, Ma, Trudinger and myself obtained the regularity under a condition denoted as A3. Loeper showed that a weaker form of the condition, denoted as A3w, is necessary. The regularity under A3w was studied by Figalli, Kim, McCann. Most recently, Li, Santambrogio and myself also studied the regularity in Monge's mass transfer problem. In this talk I will discuss the latest development in this direction.