This paper presents a trace formula for an index over the spectrum of four dimensional superconformal field theories on $ S^3 \times $ time. The index captures all information about protected short representations in 4 dimensional superconformal field theories that can be obtained purely from group theory. For the N=4 theory, the index is a function of four continuous variables. It is computed at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in $ AdS_5 \times S^5 $, showing perfect agreement at large N and small charges. The index does not reproduce the entropy of supersymmetric black holes in $ AdS_5 $, but this is not a contradiction as it differs qualitatively from the partition function over supersymmetric states of the N=4 theory. The paper also evaluates and studies the partition function over the chiral ring in the N=4 Yang Mills theory. The index is shown to be invariant under continuous deformations of the theory that preserve superconformal invariance. The paper discusses the structure of unitary representations of the conformal and superconformal algebras, and how they relate to the index. It also presents the computation of the index in N=4 Yang Mills on $ S^3 $, showing agreement with the index evaluated over the spectrum of free ten dimensional massless fields propagating on $ AdS_5 \times S^5 $. The paper also discusses the partition function over BPS states and the entropy of black holes in $ AdS_5 \times S^5 $. The index is shown to capture all group theoretically protected information about short representations in superconformal field theories. The paper concludes with a discussion of the implications of these results for the study of superconformal field theories.This paper presents a trace formula for an index over the spectrum of four dimensional superconformal field theories on $ S^3 \times $ time. The index captures all information about protected short representations in 4 dimensional superconformal field theories that can be obtained purely from group theory. For the N=4 theory, the index is a function of four continuous variables. It is computed at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in $ AdS_5 \times S^5 $, showing perfect agreement at large N and small charges. The index does not reproduce the entropy of supersymmetric black holes in $ AdS_5 $, but this is not a contradiction as it differs qualitatively from the partition function over supersymmetric states of the N=4 theory. The paper also evaluates and studies the partition function over the chiral ring in the N=4 Yang Mills theory. The index is shown to be invariant under continuous deformations of the theory that preserve superconformal invariance. The paper discusses the structure of unitary representations of the conformal and superconformal algebras, and how they relate to the index. It also presents the computation of the index in N=4 Yang Mills on $ S^3 $, showing agreement with the index evaluated over the spectrum of free ten dimensional massless fields propagating on $ AdS_5 \times S^5 $. The paper also discusses the partition function over BPS states and the entropy of black holes in $ AdS_5 \times S^5 $. The index is shown to capture all group theoretically protected information about short representations in superconformal field theories. The paper concludes with a discussion of the implications of these results for the study of superconformal field theories.