Liquid crystals are states of matter intermediate between solid crystals and normal isotropic liquids. Based on a generalization of the static liquid crystal theory,  Ericksen  and Leslie in the 1960s  proposed some constitutive equations for nematic liquid crystals in  the hydrodynamic theory of liquid crystals. The Ericksen-Leslie theory  has been verified in physics as one of the successful theories for modelling the nematic liquid crystal flow. In this talk, we will survey some results on the existence  of the solution of  the  Ericksen-Leslie system and present some new results on their Ginzburg-Landau approximations.