  We investigate the use of the embedding formalism and the Mellin transform in
the calculation of tree-level conformal correlation functions in $AdS$/CFT. We
evaluate 5- and 6-point Mellin amplitudes in $\phi^3$ theory and even a 12-pt
diagram in $\phi^4$ theory, enabling us to conjecture a set of Feynman rules
for scalar Mellin amplitudes. The general vertices are given in terms of
Lauricella generalized hypergeometric functions. We also show how to use the
same combination of Mellin transform and embedding formalism for amplitudes
involving fields with spin. The complicated tensor structures which usually
arise can be written as certain operators acting as projectors on much simpler
index structures - essentially the same ones appearing in a flat space
amplitude. Using these methods we are able to evaluate a four-point current
diagram with current exchange in Yang-Mills theory.
