This paper first reviews the shoving model for the non-Arrhenius viscosity of viscous liquids. According to this model the main contribution to the activation energy of a flow event is the energy needed for molecules to shove aside the surrounding, an energy which is proportional to the instantaneous shear modulus of the liquid. Data are presented supporting the model. It is shown that the fractional Debye–Stokes–Einstein relation, which quantitatively expresses the frequently observed decoupling of, e.g., conductivity from viscous flow, may be understood within the model. The paper goes on to review several related explanations for the non-Arrhenius viscosity. Most of these are also ‘elastic models’, i.e., they express the viscosity activation energy in terms of short-time elastic properties of the liquid. Finally, two alternative arguments for elastic models are given, a general solid-state defect argument and an Occam’s razor type argument.