  I show that under certain conditions it is possible to define consistent
irrelevant deformations of interacting conformal field theories. The
deformations are finite or have a unique running scale ("quasi-finite"). They
are made of an infinite number of lagrangian terms and a finite number of
independent parameters that renormalize coherently. The coefficients of the
irrelevant terms are determined imposing that the beta functions of the
dimensionless combinations of couplings vanish ("quasi-finiteness equations").
The expansion in powers of the energy is meaningful for energies much smaller
than an effective Planck mass. Multiple deformations can be considered also. I
study the general conditions to have non-trivial solutions. As an example, I
construct the Pauli deformation of the IR fixed point of massless non-Abelian
Yang-Mills theory with N_c colors and N_f <~ 11N_c/2 flavors and compute the
couplings of the term F^3 and the four-fermion vertices. Another interesting
application is the construction of finite chiral irrelevant deformations of N=2
and N=4 superconformal field theories. The results of this paper suggest that
power-counting non-renormalizable theories might play a role in the description
of fundamental physics.
