Hysteretic models are mathematical models capable of simulating complex nonlinear behavior (hysteresis) characterizing mechanical systems and materials used in different fields of engineering, such as aerospace, civil, and mechanical engineering. Some examples of mechanical systems and materials having hysteretic behavior are:
* materials, such as steel, reinforced concrete, wood;
* structural elements, such as steel, reinforced concrete, or wood joints;
* devices, such as seismic isolators and dampers.
Each subject that involves hysteresis has models that are specific to the subject. In addition, there are hysteretic models that capture general features of many systems with hysteresis. An example is the Preisach model of hysteresis, which represents a hysteresis nonlinearity as a linear superposition of square loops called non-ideal relays. Many complex models of hysteresis arise from the simple parallel connection, or superposition, of elementary carriers of hysteresis termed hysterons.
A simple and intuitive parametric description of various hysteresis loops may be found in the Lapshin model. Along with the smooth loops, substitution of trapezoidal, triangular or rectangular pulses instead of the harmonic functions allows piecewise-linear hysteresis loops frequently used in discrete automatics to be built in the model. There are implementations of the hysteresis loop model in Mathcad and in R programming language.
The Bouc–Wen model of hysteresis is often used to describe non-linear hysteretic systems. It was introduced by Bouc and extended by Wen, who demonstrated its versatility by producing a variety of hysteretic patterns. This model is able to capture in analytical form, a range of shapes of hysteretic cycles which match the behaviour of a wide class of hysteretical systems; therefore, given its versability and mathematical tractability, the Bouc–Wen model has quickly gained popularity and has been extended and applied to a wide variety of engineering problems, including multi-degree-of-freedom (MDOF) systems, buildings, frames, bidirectional and torsional response of hysteretic systems two- and three-dimensional continua, and soil liquefaction among others. The Bouc–Wen model and its variants/extensions have been used in applications of structural control, in particular in the modeling of the behaviour of magnetorheological dampers, base isolation devices for buildings and other kinds of damping devices; it has also been used in the modelling and analysis of structures built of reinforced concrete, steel, masonry and timber.. The most important extension of Bouc-Wen Model was carried out by Baber and Noori and later by Noori and co-workers.  That extended model, named, BWBN, can reproduce the complex shear pinching or slip-lock phenomenon that earlier model could not reproduce. The BWBN model has been widely used in a wide spectrum of applications and implementations are available in software such as OpenSees.
Hysteretic models may have a generalized displacement  as input variable and a generalized force  as output variable, or vice versa. In particular, in rate-independent hysteretic models, the output variable does not depend on the rate of variation of the input one.
Rate-independent hysteretic models can be classified into four different categories depending on the type of equation that needs to be solved to compute the output variable:
* algebraic models
* transcendental models
* differential models
* integral models