  We provide a new general setting for scalar interacting fields on the
covering of a d+1-dimensional AdS spacetime. The formalism is used at first to
construct a one-paramater family of field theories, each living on a
corresponding spacetime submanifold of AdS, which is a cylinder $R\times
S_{d-1}$. We then introduce a limiting procedure which directly produces
Luescher-Mack CFT's on the covering of the AdS asymptotic cone. Our AdS/CFT
correspondence is generally valid for interacting fields, and is illustrated by
a complete treatment of two-point functions, the case of Klein-Gordon fields
appearing as particularly simple in our context.
  We also show how the Minkowskian representation of these boundary CFT's can
be directly generated by an alternative limiting procedure involving
Minkowskian theories in horocyclic sections (nowadays called (d-1)-branes,
3-branes for AdS_5). These theories are restrictions to the brane of the
ambient AdS field theory considered. This provides a more general
correspondence between the AdS field theory and a Poincare' invariant QFT on
the brane, satisfying all the Wightman axioms. The case of two-point functions
is again studied in detail from this viewpoint as well as the CFT limit on the
boundary.
