  In theories without known Lagrangian descriptions, knowledge of the global
symmetries is often one of the few pieces of information we have at our
disposal. Gauging (part of) such global symmetries can then lead to interesting
new theories, which are usually still quite mysterious. In this work, we
describe a set of tools that can be used to explore the superconformal phases
of these theories. In particular, we describe the contribution of such
non-Lagrangian sectors to the NSVZ beta-function, and elucidate the counting of
marginal deformations. We apply our techniques to N=1 theories obtained by mass
deformations of the N=2 conformal theories recently found by Gaiotto. Because
the basic building block of these theories is a triskelion, or trivalent
vertex, we dub them "Sicilian gauge theories." We identify these N=1 theories
as compactifications of the six-dimensional A_N (2,0) theory on Riemann
surfaces with punctures and SU(2) Wilson lines. These theories include the
holographic duals of the N=1 supergravity solutions found by Maldacena and
Nunez.
