We report on the recent progress made on flows of living polymer fluids. Such fluids have been model systems for rheological research for more than twenty years and they continue to be fascinating. Like most if not all soft matter systems, living polymers under flow show a strong feedback between the structure of the fluid and that of the flow, the first influencing the second and vice versa. In our opinion, such interplay between microscopic kinetics and macroscopic kinematics has historically been mostly understood from a "structural perspective", in the tradition of physical chemistry. Nevertheless, in recent years, a more "hydrodynamical perspective" has emerged by making fruitful analogies with elastic and inertio-elastic instabilities known in solutions of regular polymers. We also underline how this new perspective constrains theoretical modelling and calls for the use of new tools of investigation.