Circadian rhythms are endogenous~24-hr oscillations usually entrained to daily environmental cycles of light/dark. Many biological processes and physiological functions including mammalian body temperature, the cell cycle, sleep/wake cycles, neurobehavioral performance, and a wide range of diseases including metabolic, cardiovascular, and psychiatric disorders are impacted by these rhythms. Circadian clocks are present within individual cells and at tissue and organismal levels as emergent properties from the interaction of cellular oscillators. Mathematical models of circadian rhythms have been proposed to provide a better understanding of and to predict aspects of this complex physiological system. These models can be used to: (a) manipulate the system in silico with specificity that cannot be easily achieved using in vivo and in vitro experimental methods and at lower cost, (b) resolve apparently contradictory empirical results, (c) generate hypotheses, (d) design new experiments, and (e) to design interventions for altering circadian rhythms. Mathematical models differ in structure, the underlying assumptions, the number of parameters and variables, and constraints on variables. Models representing circadian rhythms at different physiologic scales and in different species are reviewed to promote understanding of these models and facilitate their use. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models K E Y W O R D S biological oscillators, circadian clock, circadian rhythms, dynamic systems, mathematical modeling, statistical modeling BOX 1 CIRCADIAN MODELINGCircadian clocks are present within individual cells, and communication among multiple cells gives rise to emergent properties at the tissue level. In mammals, both the master circadian clock in the suprachiasmatic nucleus and tissuelevel peripheral clocks have major effects at the organism level on numerous key physiological functions, including sleep/wake cycles, metabolism, cardiovascular function, reproduction, immune function, neurobehavioral performance, and mood. Misalignment between the master clock and peripheral clocks within an organism, or misalignment between an organism's clocks and its external environment, has adverse physiological consequences. When circadian interventions are needed to improve physiological function, a multiscale understanding of circadian rhythmicity therefore is essential to accurately manipulate this complex oscillatory system. Mathematical modeling is an essential tool to study and analyze complex physiological systems. It has been used to provide insight into the circadian system at multiple levels (i.e., organism, multi-cellular, cellular, molecular, genetic), to design new experiments, and to manipulate and control the components of the system in silico with specificity that