I will present an elementary introduction to the regularity/uniqueness issues for the Navier-Stokes equations. Thanks to the convex integration technique, the h-principle is starting to take shape for the uniqueness problem. It might be possible that classical scaling invariant uniqueness results (Prodi-Serrin type conditions) are the only constraints for constructing non-unique solutions of the NSE. As for the regularity problem, the situation is more delicate since there are also dynamical constraints that can potentially prevent solutions from blowing up.