Whether the solution to the systems of equations in fluid mechanics, such as the Navier-Stokes and the magneto-hydrodynamics, remain smooth for all time in a three-dimensional space remains a challenging open problem. In 1962 Serrin provided a certain space-time integrability condition in a scaling-invariant norms for the weak solution to the Navier-Stokes system, which is a three-dimensional velocity vector field, to be smooth. We discuss recent developments in the research direction in effort to improve such integrability conditions so that we only have to impose the condition on only one of the three velocity vector field components, instead of all of three, as well as its extension to the magneto-hydrodynamics system. The proof crucially relies on a key identity which is a consequence of the divergence-free property, and techniques from anisotropic Littlewood-Paley theory.