The　paper　considers　the　regularity　problem　on　three-dimensional　incompressible　Navier-Stokes　equations　in　general　orthogonal　curvilinear　coordinate　systems.　We　establish　one　regularity　criteria　of　the　weak　solutions　involving　only　in　a　vorticity　component　omega(3)　and　one　a　priori　estimate　on　the　solution　that　parallel　to　H(3)u(3)parallel　to(L　infinity(0,T;Lp　(R3)))　is　bounded　for　1　＜=　p　＜=　infinity　to　three-dimensional　incompressible　Navier-Stokes　equations　in　orthogonal　curvilinear　coordinate　systems.　These　extent　greatly　the　corresponding　results　on　axisymmetric　cylindrical　flow.