Summary: Scale-invariant processes, and hereafter processes with broken versions of this symmetry, are studied by means of the Lamperti transformation, a one-to-one transformation linking stationary and self-similar processes. A general overview of the use of the transformation, and of the stationary generators it builds, is given for modelling and analysis of scale invariance. We put an emphasis on generalizations to non-strictly scale-invariant situations. The examples of discrete scale invariance and finite-size scale invariance are developed by means of the Lamperti transformation framework, and some specific examples of processes with these generalized symmetries are given.