  Recently, it has been proposed by Maldacena that large $N$ limits of certain
conformal field theories in $d$ dimensions can be described in terms of
supergravity (and string theory) on the product of $d+1$-dimensional $AdS$
space with a compact manifold. Here we elaborate on this idea and propose a
precise correspondence between conformal field theory observables and those of
supergravity: correlation functions in conformal field theory are given by the
dependence of the supergravity action on the asymptotic behavior at infinity.
In particular, dimensions of operators in conformal field theory are given by
masses of particles in supergravity. As quantitative confirmation of this
correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on
$AdS_5\times {\bf S}^5$ match with the chiral operators of $\N=4$ super
Yang-Mills theory in four dimensions. With some further assumptions, one can
deduce a Hamiltonian version of the correspondence and show that the $\N=4$
theory has a large $N$ phase transition related to the thermodynamics of $AdS$
black holes.
