  Consequences of large $N$ volume independence are examined in conformal and
confining gauge theories. In the large $N$ limit, gauge theories compactified
on $\R^{d-k} \times (S^1)^k$ are independent of the $S^1$ radii, provided the
theory has unbroken center symmetry. In particular, this implies that a large
$N$ gauge theory which, on $\R^d$, flows to an IR fixed point, retains the
infinite correlation length and other scale invariant properties of the
decompactified theory even when compactified on $\R^{d-k} \times (S^1)^k$. In
other words, finite volume effects are $1/N$ suppressed. In lattice
formulations of vector-like theories, this implies that numerical studies to
determine the boundary between confined and conformal phases may be performed
on one-site lattice models. In $N=4$ supersymmetric Yang-Mills theory, the
center symmetry realization is a matter of choice: the theory on
$\R^{4-k}\times (S^1)^k$ has a moduli space which contains points with all
possible realizations of center symmetry. Large $N$ QCD with massive adjoint
fermions and one or two compactified dimensions has a rich phase structure with
an infinite number of phase transitions coalescing in the zero radius limit.
