  We revisit rescaling methods for nonlinear elliptic and parabolic problems
and show that, by suitable modifications, they may be used for nonlinearities
that are not scale invariant even asymptotically and whose behavior can be
quite far from power like.
  In this enlarged framework, by adapting the doubling-rescaling method from
[37, 38], we show that the equivalence found there between universal estimates
and Liouville theorems remains valid. In the parabolic case we also prove a
Liouville type theorem for a rather large class of non scale invariant
nonlinearities. This leads to a number of new results for non scale invariant
elliptic and parabolic problems, concerning space or space-time singularity
estimates, initial and final blow-up rates, universal and a priori bounds for
global solutions, and decay rates in space and/or time.
  We illustrate our approach by a number of examples, which in turn give
indication about the optimality of the estimates and of the assumptions.
