We identify and characterize a class of term structure models where bond yields are quadratic functions of the state vector.We label this class the quadratic class and aim to lay a solid theoretical foundation for its future empirical application.We consider asset pricing in general and derivative pricing in particular under the quadratic class.We provide two general transform methods in pricing a wide variety of fixed income derivatives in closed or semi-closed form.We further illustrate how the quadratic model and the transform methods can be applied to more general settings.