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In this talk,we will first present a unified proof on partial regularity of suitable weak solutions to non-stationary and stationary Navier-Stokes equations proved by previous authors such as L.Caffarelli,R.Kohn and L.Nirenberg,M.Struwe,H.Dong and D.Du,H.Dong and R.Strain.Particularly,we obtain the partial regularity of the suitable weak solutions to the 4D time-dependent Navier-Stokes equations.Our proof relies on the De Giorgi iteration recently developed by A.Vasseur and some elementary observation of these equations.Then we show some anisotropic regularity conditions for the suitable weak solutions to the 3D Navier-Stokes equations.Finnally,we are concerned with possible time singular points and eventual regularity of weak solutions to the Navier-Stokes equations with fractional dissipation $(-Delta)^{alpha}$.My co-authors are Professor Quansen Jiu and Dr.Gang Wu.