  Conformal invariance is spontaneously broken in many physical systems leading
to the appearance of a single massless Goldstone mode in the spectrum, the
dilaton. The dilaton soft limit is shown to generically encode the action of
both the dilatation and the special conformal transformation on observables.
For massive on-shell legs these take the form of sub-leading soft theorems. At
loop level, we show how anomalous dimensions for coupling constants and fields
can be included. We illustrate the general analysis with a variety of formal
and phenomenological applications.
