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In this talk I will try to give an introduction to implicit constitutive theories. In particular, I will focus on strain-limiting models introduced by Rajagopal, where a nonlinear relationship is assumed among the linearized strain, strain rate and the Cauchy stress. I will then talk about a recent joint work with H. A. Erbay where we investigate traveling wave solutions of a nonlinear differential equation describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We establish conditions for the existence of such solutions, and find those solutions explicitly, implicitly, or numerically, for various forms of the nonlinear constitutive relation.