Feedback is used to control systems whose open-loop behaviour is uncertain. Over the last twenty years a mature theory of robust control has been developed for linear multivariable systems in continuous time. But most practical control systems have constraints such as saturation limits on the actuators, which render the closed-loop nonlinear. Most of the modern controllers are also implemented digitally using computers.The study of this research is divided in two directions: the stability analysis of discrete-time Lur'e systems and the synthesis of static discrete-time anti-windup schemes. With respect to stability analysis, the main contributions of this thesis are the derivations of new LMI-based stability criteria for the discrete-time Lur'e systems with monotonic, slope-restricted nonlinearities via the Lyapunov method. The criteria provide convex stability conditions via LMIs, which can be efficiently computed via convex optimization methods. They are also extended to the general case that includes the non-diagonal MIMO nonlinearities. The importance of extending them to the general case is that it can eventually be applied to the stability analysis of several optimization-based controllers such as an input-constrainedmodel predictive control (MPC), which is inherently discrete. With respect to synthesis, the contribution is the convex formulation of a static discrete-time anti-windup scheme via one of the Jury-Lee criteria (a discrete-time counterpart of Popov criterion), which was previously conjectured to be unachievable. The result is also in the form of LMI, and is extended to several existing static anti-windup schemes with open-loop stable plants.