This paper is concerned with the design of gain-scheduled controllers with guaranteed H infinity performance for a class of linear parameter-varying (LPV) plants. Here the plant state-space matrices are assumed to depend affinely on a vector theta of time-varying real parameters. Assuming real-time measurement of these parameters, they can be fed to the controller to optimize the performance and robustness of the closed-loop system. The resulting controller is time-varying and automatically ’gain-scheduled’ along parameter trajectories. Based on the notion of quadratic H infinity performance, solvability conditions are obtained for continuous- and discrete-time systems. In both cases the synthesis problem reduces to solving a system of linear matrix inequalities (LMIs). The main benefit of this approach is to bypass most difficulties associated with more classical schemes such as gain-interpolation or gain-scheduling techniques. The methodology presented in this paper is applied to the gain scheduling of a missile autopilot. The missile has a large operating range and high angles of attack. The difficulty of the problem is reinforced by tight performance requirements as well as the presence of flexible modes that limit the control bandwidth.