  We provide dramatic evidence that `Mellin space' is the natural home for
correlation functions in CFTs with weakly coupled bulk duals. In Mellin space,
CFT correlators have poles corresponding to an OPE decomposition into `left'
and `right' sub-correlators, in direct analogy with the factorization channels
of scattering amplitudes. In the regime where these correlators can be computed
by tree level Witten diagrams in AdS, we derive an explicit formula for the
residues of Mellin amplitudes at the corresponding factorization poles, and we
use the conformal Casimir to show that these amplitudes obey algebraic finite
difference equations. By analyzing the recursive structure of our factorization
formula we obtain simple diagrammatic rules for the construction of Mellin
amplitudes corresponding to tree-level Witten diagrams in any bulk scalar
theory. We prove the diagrammatic rules using our finite difference equations.
Finally, we show that our factorization formula and our diagrammatic rules
morph into the flat space S-Matrix of the bulk theory, reproducing the usual
Feynman rules, when we take the flat space limit of AdS/CFT. Throughout we
emphasize a deep analogy with the properties of flat space scattering
amplitudes in momentum space, which suggests that the Mellin amplitude may
provide a holographic definition of the flat space S-Matrix.
