Conservation laws with nonlocal fluxes have recently drawn considerable attention owing to their applications to several engineering problems, like models of vehicular and pedestrian traffic. They consist of conservation laws where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a (classical) conservation law. In this talk I will overview recent progress on the rigorous justification of this nonlocal-to-local limit. I will mention counter-examples showing that, in general, the solutions of the nonlocal problems do not converge to the entropy admissible solution of the conservation law. On the other hand, the nonlocal-to-local limit have been recently justified, under different assumptions, in the case of anisotropic convolution kernels, which are natural in view of applications to models of vehicular traffic. The talk will be based on joint works with Maria Colombo, Gianluca Crippa and Elio Marconi.