Since 1934 when Leray proved the global existence of finite energy weak solutions to 3D incompressible Navier-Stokes equations, the regularity and uniqueness of Leray weak solutions has been core in the mathematical theory of incompressible fluid mechanics. This problem is also closely related to the Millionaires problem proposed by Clay Institute, namely, the global regularity or singularity to finite energy solutions of 3D incompressible Navier-Stokes system. The goal of this article is first to introduce the problem and then to survey some progresses on this problem.In Section 1 , following Fefferman, we first present the Millionaires problem concerning 3D incompressible Navier-Stokes equations.We also present the result of Leray on the global existence of finite energy weak solutions to the system. We conclude the introduction byremarking that the reason why we cannot solve the global regularity of 3D Navier-Stokes is that the system is energy super-critical. Whereas the energy lawis so far the only conservation law which we can find for 3D Navier-Stokes system.The goal of Section 2 is to present the main approaches toward the regularity of 3D Navier-Stokes system, namely, partial regularityof suitable weak solutions to 3D Navier-Stokes system, conditional regularity of Leray solutions, and well-posedness of 3D Navier-Stokes system with initial datain the critical spaces.Finally, in the last section, we comment about the possible singularity of 3D Navier-Stokes system.