Controversy about the frame indifference principle, the concept of non-local continuum field theories, local constitutive formulations, differential constitutive equations of linear viscoelasticity, Oldroyd, K-BKZ, FENE (Finitely Extensible Non-linear Elastic) class of constitutive equations, Smoluchowski and Fokker–Planck diffusion equations, constant stretch history flows, fading memory and nested integral representations of the stress, order fluids of the integral and differential type, constitutive formulations consistent with thermodynamics, maximization of the rate of dissipation in formulating thermodynamics compatible constitutive structures, Burgers equation which is finding a gradually widening niche in applications, minimum free energy and maximum recoverable work in the case of linearized viscoelastic constitutive structures, implicit constitutive theories, which define the stress field when the viscosity depends for instance on the constitutively undetermined pressure field, and which have found new focus in applications such as elastohydrodynamic lubrication are discussed and progress made is summarized. Canonical forms of Maxwell-like constitutive differential equations and single integral constitutive equations are presented and commented on together with the Hadamard and dissipative type of instabilities they may be subject to.