The regularity property of optimal transport problems with a general cost function is a highly non-linear problem. It was studied by many researchers, but one breakthrough was made by X. Ma, N. Trudinger, and X. Wang in 2005 where they used an assumption on a 4-tensor defined using the cost function. Since then, the meaning of the condition was discovered by other researchers in different statements independently. In this talk, we discuss the condition and its other expressions discovered by other researchers and I present another formulation obtained in my recent work.