  We study the chiral rings in N=2 and N=4 superconformal algebras. The chiral
primary states of N=2 superconformal algebras realized over hermitian triple
systems are given. Their coset spaces G/H are hermitian symmetric which can be
compact or non-compact. In the non-compact case, under the requirement of
unitarity of the representations of G we find an infinite set of chiral primary
states associated with the holomorphic discrete series representations of G.
Further requirement of the unitarity of the corresponding N=2 module truncates
this infinite set to a finite subset. The chiral primary states of the N=2
superconformal algebras realized over Freudenthal triple systems are also
studied. These algebras have the special property that they admit an extension
to N=4 superconformal algebras with the gauge group SU(2)XSU(2)XU(1). We
generalize the concept of the chiral rings to N=4 superconformal algebras. We
find four different rings associated with each sector (left or right moving).
We also show that our analysis yields all the possible rings of N=4
superconformal algebras.
