Motivated by recently explored examples, we undertake a systematic study of conformal invariance in one-dimensional sigma models where an isometry group has been gauged. Perhaps surprisingly, we uncover classes of sigma models which are only scale invariant in their ungauged form and become fully conformally invariant only after gauging. In these cases the target space of the gauged sigma model satisfies a deformation of the well-known conformal geometry constraints. We consider bosonic models as well as their N = 1,2,4 supersymmetric extensions. We solve the quantum ordering ambiguities in implementing (super-) conformal symmetry on the physical Hilbert space. Examples of our general results are furnished by the D(2,1;0)-invariant Coulomb branch quiver models relevant for black hole physics.