  In this paper we investigate the three dimensional general Ericksen-Leslie
(E--L) system with Ginzburg-Landau type approximation modeling nematic liquid
crystal flows. First, by overcoming the difficulties from lack of maximum
principle for the director equation and high order nonlinearities for the
stress tensor, we prove existence of global-in-time weak solutions under
physically meaningful boundary conditions and suitable assumptions on the
Leslie coefficients, which ensures that the total energy of the E--L system is
dissipated. Moreover, for the E--L system with periodic boundary conditions, we
prove the local well-posedness of classical solutions under the so-called
Parodi's relation and establish a blow-up criterion in terms of the temporal
integral of both the maximum norm of the curl of the velocity field and the
maximum norm of the gradient of the liquid crystal director field.
