We consider sufficient conditions for regularity of Leray-Hopf solutions of the Navier-Stokes equation. By a result of Neustupa and Panel, a Leray-Hopf weak solution is regular provided a single component of the velocity is bounded. In this talk we will survey existing and present new results on one component and one direction regularity. We will also show global regularity for a class of solutions of the Navier-Stokes equation in thin domains. This is a joint work with M. Ziane.