Hysteresis is a phenomenon which is widely observed in a variety of physical systems. It introduces a nonlinear and multivalued behavior in systems, making their modeling and control problematic. This thesis underlines the significance of dynamic hysteresis modeling from the perspectives of analysis and control. Toward that end, a widely accepted definition of hysteresis is adopted and some important properties of hysteresis are presented. Five general hysteresis models are discussed, along with some damping and friction models. Their properties are compared and contrasted. The Duhem model is shown to be a versatile dynamic hysteresis model, and it is adapted to two distinct physical systems. First, the evolution of dynamic hysteresis modeling of harmonic drive is studied, and a new dynamic model, based on Duhem model, is developed. It is more accurate than previous models and is used to prove, via the method of describing functions, that PID regulation control of harmonic drive can cause a limit cycle due to hysteresis. Second, a dynamic hysteresis model, based on Duhem model, is proposed for a shape memory alloy actuator, which yields a complete dynamic model of the actuator, linking its temperature, strain and electrical resistance together. Therefore, this thesis provides a foundation for dynamic hysteresis modeling in engineering systems and brings out the salient features of dynamic hysteresis modeling from the perspectives of analysis and control.