F riction is a dynamic phenomenon of widespread importance, and the associated literature is vast; overviews are given in [1]- [5]. Friction can be viewed as an emergent, macroscopic phenomenon arising from molecular interaction. Consequently, both physical (physics-based) and empirical (experimentbased) models have been studied [2], [6]- [14]. Estimation and control methods are available for applications involving friction [15]-[18]; however, these topics are beyond the scope of this article.Friction models distinguish between presliding friction and sliding friction. Presliding or micro-slip friction refers to the friction forces that occur when the relative displacement between two contacting surfaces is microscopic, that is, on the order of the asperities (roughness features) on the surfaces. Sliding friction refers to the friction forces that arise when the relative displacement is macroscopic. Understanding presliding friction is useful for high precision motion control applications. For example, hysteresis can occur between the presliding friction force input and the displacement output [7], [11], [12].From a mathematical point of view, friction modeling is challenging since these models often involve nonsmooth dynamics. For example, the most widely used dry friction model, namely, Coulomb friction, is discontinuous. Additional discontinuous dry friction models are studied in [19]. Some friction models are continuous but have nonLipschitzian dynamics, which is a necessary condition for finite-settling-time behavior and the associated lack of time-reversibility [20], [21]. Table 1 classifies the properties of some widely used friction models.Hysteresis is the result of multistability, which refers to the existence of multiple attracting equilibria [22]-[24]. Multistability implies that hysteresis is a quasi-static phenomenon in the sense that the hysteresis map is the limit of a sequence of periodic dynamic input-output maps as the period of the input increases without bound. In both presliding and sliding friction models, there exist multiple equilibria corresponding to states that correspond to constant friction forces under constant displacement or velocity.In this article we examine several classical friction models from a hysteresis modeling point of view and study the