Some new sufficient conditions for determining global asymptotic stability of an equilibrium point under control saturation are presented. For an open-loop stable system, we show that there exists a class of linear time invariant state feedback matrices which globally stablizes an equilibrium under control saturation. In addition, we show that for open-loop unstable plants, an equilibrium can never be globally stabilized using a certain class of saturated linear time invariant state feedback. Finally, we extend the stability results to nonlinear systems whose nonlinearities are globally Lipschitz.