An absolute stabilization framework based on classical notions of absolute stability is developed to design anti-windup and guaranteed relative stability margin controllers for systems with input time-varying nonlinearities. The approach is directly applicable to systems with saturating actuators and provides fixed-order (i.e., full-and reduced-order) dynamic controllers with a guaranteed domain of attraction. The proposed framework can also be used to design fixed-order controllers with guaranteed gain and phase margins for multiloop feedback systems. A numerical algorithm based on a discrete homotopy is developed to design absolute stabilization controllers.