For a class of linear dynamical systems with constant unknown parameters an ℒ 1 adaptive control framework is developed that provides stable adaptation in the presence of input constraints. For open-loop stable systems the results are global. For open-loop unstable systems the derivation of the non-empty positive invariant set is cast into a linear matrix inequality (LMI) framework, which can be solved numerically via an appropriate toolbox of convex optimization. Performance bounds of the ℒ 1 adaptive closed-loop system are analyzed to capture the impact of the input saturation. A simulation is given to verify the theoretical statements.