Although actuator saturation is a common nonlinear problem in practical control systems, the condition under which compensators exist that globally stabilize systems subject to actuator saturation, is still not well understood. In this paper, it is shown that actuator saturation compensators exist that globally stabilize systems containing up to one integrator. For systems with two integrators, it is established that the compensated system can only be Lagrange stable. The stability result presented in this paper can also be extended to analyse the stability of systems using existing compensation schemes, thus providing a much needed justification for the use of these schemes in practice. The results presented here are illustrated by examples.