We construct classes of ${\cal N}=1$ superconformal theories elements of
which are labeled by punctured Riemann surfaces. Degenerations of the surfaces
correspond, in some cases, to weak coupling limits. Different classes are
labeled by two integers (N,k). The k=1 case coincides with A_{N-1} ${\cal N}=2$
theories of class S and simple examples of theories with k>1 are Z_k orbifolds
of some of the A_{N-1} class S theories. For the space of ${\cal N}=1$ theories
to be complete in an appropriate sense we find it necessary to conjecture
existence of new ${\cal N}=1$ strongly coupled SCFTs. These SCFTs when coupled
to additional matter can be related by dualities to gauge theories. We discuss
in detail the A_1 case with k=2 using the supersymmetric index as our analysis
tool. The index of theories in classes with k>1 can be constructed using
eigenfunctions of elliptic quantum mechanical models generalizing the
Ruijsenaars-Schneider integrable model. When the elliptic curve of the model
degenerates these eigenfunctions become polynomials with coefficients being
algebraic expressions in fugacities, generalizing the Macdonald polynomials
with rational coefficients appearing when k=1.