Whether the 3D incompressible Euler equation can develop a finite time singularity from smooth initial data has been an outstanding open problem for over a century. Here we review some existing computational and theoretical work on possible finite blow-up of the 3D Euler equation. We show that there is a sharp relationship between the geometric properties of the vortex filament and the maximum vortex stretching. By exploring this local geometric property of the vorticity field, we prove the global regularity of the 3D incompressible Euler equations provided that the unit vorticity vector and the velocity field satisfy certain localized regularity conditions. Further, we apply this localized non-blowup criterion to re-examine some of the most well-known numerical evidences in which a finite-time blowup of the 3D Euler equations has been alleged. Our well-resolved large scale computations show that the local geometric regularity of the vorticity vectors leads to tremendous dynamic depletion of the nonlinear vortex stretching term, which prevents the finite time blowup of the 3D Euler equations. This confirms our localized non-blowup theory. Finally, we present a new class of solutions for the 3D Euler and Navier-Stokes equations, which exhibit very interesting dynamic growth property. By exploiting the special nonlinear structure of the equations, we can prove nonlinear stability and the global regularity of this class of solutions.