  In this work we propose a way to promote the anomalous axial U(1)
transformations to exact non-invertible U(1) symmetries. We discuss the
procedure of coupling the non-invertible symmetry to a (dynamical or
background) gauge field. We show that as part of the gauging procedure, certain
constraints are imposed to make the gauging consistent. The constraints emerge
naturally from the form of the non-invertible U(1) conserved current. In the
case of dynamical gauging, this results in new type of gauge theories we call
non-invertible gauge theories: These are gauge theories with additional
constraints that cancel the would-be gauge anomalies. By coupling to background
gauge fields, we can discuss 't-Hooft anomalies of non-invertible symmetries.
We show in an example that the matching conditions hold but they are realized
in an unconventional way. Turning on non-trivial background for the
non-invertible gauge field changes the vacuum even when the symmetry is not
broken and the background is very weak. The anomalies are then matched by the
appearance of solitons in the new vacuum.
