  In this paper we discuss candidate superconformal N=2 gauge theories that
realize the AdS/CFT correspondence with M--theory compactified on the
homogeneous Sasakian 7-manifolds M^7 that were classified long ago. In
particular we focus on the two cases M^7=Q^{1,1,1} and M^7=M^{1,1,1}, for the
latter the Kaluza Klein spectrum being completely known. We show how the toric
description of M^7 suggests the gauge group and the supersingleton fields. The
conformal dimensions of the latter can be independently calculated by
comparison with the mass of baryonic operators that correspond to 5-branes
wrapped on supersymmetric 5-cycles and are charged with respect to the Betti
multiplets. The entire Kaluza Klein spectrum of short multiplets agrees with
these dimensions. Furthermore, the metric cone over the Sasakian manifold is a
conifold algebraically embedded in some C^p. The ring of chiral primary fields
is defined as the coordinate ring of C^p modded by the ideal generated by the
embedding equations; this ideal has a nice characterization by means of
representation theory. The entire Kaluza Klein spectrum is explained in terms
of these vanishing relations. We give the superfield interpretation of all
short multiplets and we point out the existence of many long multiplets with
rational protected dimensions, whose presence and pattern seem to be universal
in all compactifications.
