Appears in collection : Advances in Nonlinear Analysis and Nonlinear Waves, a conference in honor of Frank Merle

The incompressible Navier-Stokes and Euler equations are fundamental PDEs in mathematical fluid dynamics and their well-posedness theory is nowadays largely open. The past decade has seen a surprising and remarkable progress, through various different attempts, in describing some non-unique solutions of these PDEs. The talk will survey some of the recent contributions in this direction, including works in collaboration with Albritton and Brué which show that Leray-Hopf solutions of the forced Navier-Stokes equations are not unique.