I will start with some comments about the theoretical and experimental status of conformal invariance in critical phenomena, first proposed by A. Polyakov in 1970. I will then focus on a couple of experimentally significant models of statistical physics (ferromagnets in presence of strong dipolar forces, fluctuating membranes). Surprisingly, their fixed points violate the common lore that scale invariance implies conformal invariance. How do these models manage to do so? Why is the virial current not renormalized? A shift symmetry will help to answer these questions.