  The question of spontaneous apparition of singularity in the 3D
incompressible Euler equations is one of the most important and challenging
open problems in mathematical fluid mechanics. In this survey article we review
some of recent approaches to the problem. We first review Kato's classical
local well-posedness result in the Sobolev space and derive the celebrated
Beale-Kato-Majda criterion for finite time blow-up. Then, we discuss recent
refinements of the criterion as well as geometric type of theorems on the
sufficiency condition for the regularity of solutions. After that we review
results excluding some of the scenarios leading to finite time singularities.
We also survey studies of various simplified model problems.
  A dichotomy type of result between the finite time blow-up and the global in
time regular dynamics is presented, and a spectral dynamics approach to study
local in time behaviors of the enstrophy is also reviewed. Finally, progresses
on the problem of optimal regularity for solutions to have conserved quantities
are presented.
