This paper presents a proof of stability and performance bounds of the ℒ1 adaptive control architecture in the presence of input constraints. We prove that by appropriate modification of the state predictor, which is used for the definition of the error signal in the adaptive laws, the stability and the performance bounds of the ℒ1 adaptive controller can be quantified within an appropriately defined do- main of attraction. While for open-loop stable system the results are global, for open-loop unstable systems the domain of attraction is shown to depend upon the conservative knowledge of the uncertain parameters, the choice of the filter and the amplitude constraint of the actuator. The performance bounds can be sys- tematically improved by increasing the rate of adaptation. Simulations verify the theoretical findings.