In this talk, I will review briefly the general Ericksen-Leslie system modeling the hydrodynamic motion of the nematic liquid crystals proposed by Ericksen and Leslie back in 1960’s. I will focus on the mathematical analysis of a simplified version of the Ericksen-Leslie system, proposed by Lin, which is a strong coupling between the Navier Stokes equation and the transported heat flow of harmonic maps into the two sphere. I will then present some recent results on the global existence of Leray-Hopf type weak solutions in dimension two, and several well-posedness results for small initial data in various function spaces in dimension three. It is based on joint works with Fanghua Lin, Junyu Lin, Tao Huang, and Jay Hineman.