Summary Increasing flooding-solution viscosity with polymers provides a favorable mobility ratio compared with brine flooding and hence improves volumetric sweep efficiency. Flooding with a polymer solution exhibiting elastic properties has been reported to increase displacement efficiency, resulting in a sustained doubling of the recovery enhancement compared with the use of conventional viscous-polymer flooding (Wang et al. 2011). Flooding with viscoelastic-polymer solutions is claimed also to increase recovery more than expected from changes in capillary number alone (Wang et al. 2010). This increase in displacement efficiency by viscoelastic polymers is reported to occur because of changes in the steady-state-flow profile and enhancements in oil stripping and thread formation. However, within the industry there are doubts that a genuine effect is observed, or that improvements in displacement efficiency occur with field-applicable flow regimes (Vermolen et al. 2014). In this study, we demonstrate that flooding with viscoelastic-polymer solutions can indeed increase recovery more than expected from changes in capillary number. We show a mechanism of fluctuations in flow at low Reynolds number by which viscoelastic-polymer solutions provide improvements in displacement efficiency. The mechanism, known as elastic turbulence, is an effect previously unrecognized in this context. We demonstrate that the effect may be obtained at field-relevant flow rates. Furthermore, this underlying mechanism explains both the enhanced capillary-desaturation curves and the observation of apparent flow thickening (Delshad et al. 2008; Seright et al. 2011) for these viscoelastic solutions in porous media. The work contrasts experiments on flow and recovery by use of viscous and viscoelastic-polymer solutions. The circumstances under which viscoelasticity is beneficial are demonstrated. The findings are applicable to the design of formulations for enhanced oil recovery (EOR) by polymer flooding. A combination of coreflooding, micromodel flow, and rheometric studies is presented. The results include single-phase and multiphase floods in sandstone cores. Polymer solutions are viscoelastic [partially hydrolyzed polyacrylamide (HPAM)] or viscous (xanthan). The effects of molecular weight, flow rate, and concentration of the HPAMs are described. The data lead us to suggest a mechanism that may be used to explain the observations of improved displacement efficiency and why the improvement is not seen for all viscoelastic-polymer floods.