  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 $\mathcal{N} = 1,2,4$
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.
