  We demonstrate how one can construct renormalizable perturbative expansion in
formally nonrenormalizable higher dimensional field theories. It is based on
$1/N_f$-expansion and results in a logarithmically divergent perturbation
theory in arbitrary high space-time dimension. First, we consider a simple
example of $N$-component scalar filed theory and then extend this approach to
Abelian and non-Abelian gauge theories with $N_f$ fermions. In the latter case,
due to self-interaction of non-Abelian fields the proposed recipe requires some
modification which, however, does not change the main results. The resulting
effective coupling is dimensionless and is running in accordance with the usual
RG equations. The corresponding beta function is calculated in the leading
order and is nonpolynomial in effective coupling. It exhibits either UV
asymptotically free or IR free behaviour depending on the dimension of
space-time. The original dimensionful coupling plays a role of a mass and is
also logarithmically renormalized. We analyze also the analytical properties of
a resulting theory and demonstrate that in general it acquires several ghost
states with negative and/or complex masses. In the former case, the ghost state
can be removed by a proper choice of the coupling. As for the states with
complex conjugated masses, their contribution to physical amplitudes cancels so
that the theory appears to be unitary.
