Abstract: In pipe and channel flows turbulence generally arises if velocities and hence inertial forces are sufficiently large. In striking contrast, viscoelastic fluids can exhibit disordered motion even at vanishing inertia. Intermediate between these cases, a novel state of chaotic motion, ‘elasto-inertial turbulence’ (EIT), has been observed in a narrow Reynolds number interval. I will here discuss the relation between EIT and polymer drag reduction, in particular its relation to the so called ‘maximum drag reduction asymptote’ at high Reynolds numbers. I will moreover show that in experiments the onset of EIT can be traced to Reynolds numbers of order one. In this limit the flow structures closely resemble those predicted by a recently discovered linear instability for viscoelastic fluids.