The near-field dynamics in the moving contact-line problem described in the framework of several alternative theories is examined in the general case of finite capillary numbers against the most basic criteria to be satisfied by a mathematical model. It is shown that both types of so-called 'slip models' invariably fail the criteria: they lead to solutions either with a pressure singularity at the contact line or with the flow kinematics qualitatively different from that observed in experiments, or both. The same criteria applied to an early developed theory of dynamic wetting as an interface formation process are shown to be satisfied: the model leads to a singularity-free solution with the correct 'rolling' kinematics. The flow exhibits no spurious low-velocity region near the contact line which is unavoidable in all 'slip models'. The pressure at the contact line remains finite, and the dynamic contact angle, being part of the solution, depends on the flow field and not only on the contact-line speed. This last feature accounts for the effect of 'hydrodynamic assist of dynamic wetting' reported in recent experiments. The implications of the results for numerical algorithms incorporating different models for the moving contact line are discussed. (c) 2006 Elsevier B.V. All rights reserved.