  Techniques based on $n$-particle irreducible effective actions can be used to
study systems where perturbation theory does not apply. The main advantage,
relative to other non-perturbative continuum methods, is that the hierarchy of
integral equations that must be solved truncates at the level of the action,
and no additional approximations are needed. The main problem with the method
is renormalization, which until now could only be done at the lowest ($n$=2)
level. In this paper we show how to obtain renormalized results from an
$n$-particle irreducible effective action at any order. We consider a symmetric
scalar theory with quartic coupling in four dimensions and show that the 4 loop
4-particle-irreducible calculation can be renormalized using a renormalization
group method. The calculation involves one bare mass and one bare coupling
constant which are introduced at the level of the Lagrangian, and cannot be
done using any known method by introducing counterterms.
