  We study the duality group of $\hat{A}_{n-1}$ quiver gauge theories,
primarily using their M5-brane construction. For $\mathcal{N}=2$ supersymmetry,
this duality group was first noted by Witten to be the mapping class group of a
torus with $n$ punctures. We find that it is a certain quotient of this group
that acts faithfully on gauge couplings. This quotient group contains the
affine Weyl group of $\hat{A}_{n-1}$, $\mathbb{Z}_n$ and $SL(2,\mathbb{Z})$. In
fact there are $n$ non-commuting $SL(2,\mathbb{Z})$ subgroups, related to each
other by conjugation using the $\mathbb{Z}_n$. When supersymmetry is broken to
$\mathcal{N}=1$ by masses for the adjoint chiral superfields, an RG flow ensues
which is believed to terminate at a CFT in the infrared. We find the explicit
action of this duality group for small values of the adjoint masses, paying
special attention to when the sum of the masses is non-zero. In the
$\mathcal{N}=1$ CFT, Seiberg duality acts non-trivially on both gauge couplings
and superpotential couplings and we interpret this duality as inherited from
the $\mathcal{N}=2$ parent theory. We conjecture the action of S-duality in the
CFT based on our results for small mass deformations. We also consider
non-conformal deformations of these $\mathcal{N}=1$ theories. The cascading RG
flows that ensue are a one-parameter generalization of those found by Klebanov
and Strassler and by Cachazo {\it et. al.}. The universality exhibited by these
flows is shown to be a simple consequence of paths generated by the action of
the affine Weyl group.
