This paper presents a fundamental study of the connection between continuous-and discrete-time systems. Provided is a deΓΏnition for discrete-time models, that is discrete-time systems with a continuous-time counterpart, whose order can be higher than that of the continuous-time system. This deΓΏnition is based on a comparison in a certain sense on the time responses of continuous-and discrete-time systems. A theorem is presented for relating the higher-order discrete-time models to their continuous-time counterparts, which is an extension of a previous theorem for models with order equal to that of the continuous-time system. State-space forms are derived for models obtained through the use of a certain class of hold elements and through the use of mapping models, and these discrete-time systems are shown to be valid according to the deΓΏnition. Special cases are models obtained using ΓΏrst-order and slewer hold devices, whose convergence to a continuous-time counterpart has not been shown mathematically before, and mapping models corresponding to two-step linear multi-step methods, which have not previously presented in the state-space form. The derived state-space forms provide a convenient way to implement these models for purposes of analysis, design, and implementation of discrete-time systems and ΓΏnds applications in such areas as digital signal processing, digital simulation, and digital control.