I will present some recent results on the Serrin-type conditional regularity of the Navier-Stokes equations. Basically, if one component of the weak solution of the Navier-Stokes equation belongs to a Serrin type space of regularity then the weak solution is regular and is unique. The second part of the talk is devoted to the primitive equations of the ocean with the Dirichlet boundary condition for which we prove the global regularity. This is a joint work with I. Kukavica.