I will present a general solution to the problem of determining all
S-dual descriptions for a specific (but very rich) class of N=1 SCFTs. These
SCFTs are indexed by decorated toric diagrams, and can be engineered in string
theory by probing orientifolds of isolated toric singularities with D3 branes.
The S-dual phases are described by quiver gauge theories coupled to specific
types of conformal matter which I will describe explicitly.