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The initial-boundary value problem for the density-dependent flow of nematic crystals is studied in a 2-D bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is proved for the global strong solution with the large initial velocity u0 and small ∇d0. We also give a regularity criterion ∇d∈Lp(0,T;Lq(Ω))  (2/q)+(2/p)=1, 2<q≤∞ of the problem with the Dirichlet boundary condition u=0, d=d0 on ∂Ω.