Small concentrations of long-chain polymers in a Newtonian solvent can give rise to interesting new behaviour. Most notably, `Elastic’ turbulence, which was discovered in 2000 in curvilinear geometries, can exist at vanishingly small Reynolds numbers Re while an apparently-different `Elasto-inertial’ turbulence, discovered in 2013 in straight geometries, can exist at finite Re and large Weissenberg numbers. I’ll discuss recent progress in trying to understand how these states come about and whether they are related.