What does it mean for a theory to be renormalizable

[Lemaho]

Dear all,

I have a simple question with a possibly long answer.
What does it mean for a theory to be renormalizable. And how is it related to the scaling dimension?

 

[tom.stoer]

Renormalizable means that either the theory contains only finite expressions or that infinities can be absorbed by redefining finitly many quantities. A theory is not renormalizable if you need infinitly many quantities to absorb the infinities.

Usually renormalizable is used in the sense of "perturbatively renormalizable", but this is misleading. It can very well be that a theory seems to be unrenormalizable when treated perturbatively, but this behavior can simply be due to the (unjustified) perturbation expansion and may vanish when treated non-perturbatively.