In this talk we will discuss some recent progress concerning the Navier-Stokes and Euler equations of incompressible fluid. In particular, issues concerning the lack of uniqueness using the convex integration machinery and their physical relevance. Moreover, we will discuss the Onsager conjecture for the preservation of energy in Euler equations, and its possible extension for the preservation of generalized entropy in general conservation laws and inviscid systems. In addition, we will present a blow-up criterion for the 3D Euler equations based on a class of inviscid regularization for these equations and the effect of physical boundaries on the potential formation of singularity.