Theories and attendant methodologies developed independently of thermodynamic considerations and set within a thermodynamic framework to derive theological constitutive equations for viscoelastic fluids have been reviewed in their historical context. The stability of Maxwell-like differential and single integral type constitutive formulations in current use and their relationship to experimentally observed physical instabilities are reviewed in particular in the light of inherent Hadamard and dissipative type of instabilities they may be subject to as a consequence of defective constitutive formulations. The state of the art in predicting the longitudinal field, the pressure drop and the friction factors for the flow of generalized Newtonian and viscoelastic fluids in circular and non-circular straight tubes is reviewed. (C) 2011 Elsevier Ltd. All rights reserved.