The commonly employed mathematical functions in constitutive models, such as the strain hardening/softening model, strain-rate hardening factor, and temperature-softening factor, are reviewed, and their prediction characteristics are illustrated. The results may assist one (i) to better understand the behavior of the constitutive model that employs a given mathematical function; (ii) to find the reason for deficiencies, if any, of an existing constitutive model; (iii) to avoid employing an inappropriate mathematical function in future constitutive models. This study subsequently illustrates the flow stress description characteristics of twelve constitutive models at wide strain rates (from 10−6 to 106 s−1) and temperatures (from absolute to melting temperatures) using the material parameters presented in the original studies. The phenomenological models considered herein include the Johnson–Cook, Shin–Kim, Lin–Wagoner, Sung–Kim–Wagoner, Khan–Huang–Liang, and Rusinek–Klepaczko models. The physically based models considered are the Zerilli–Armstrong, Voyiadjis–Abed, Testa et al., Steinberg et al., Preston–Tonks–Wallace, and Follansbee–Kocks models. The illustrations of the behavior of the foregoing constitutive models may be informative in (i) selecting an appropriate constitutive model; (ii) understanding and interpreting simulation results obtained using a given constitutive model; (iii) finding a reference material to develop future constitutive models.