  We give a detailed account of the methods introduced in [1] to calculate
holographic four-point correlators in IIB supergravity on $AdS_5 \times S^5$.
Our approach relies entirely on general consistency conditions and maximal
supersymmetry. We discuss two related methods, one in position space and the
other in Mellin space. The position space method is based on the observation
that the holographic four-point correlators of one-half BPS single-trace
operators can be written as finite sums of contact Witten diagrams. We
demonstrate in several examples that imposing the superconformal Ward identity
is sufficient to fix the parameters of this ansatz uniquely, avoiding the need
for a detailed knowledge of the supergravity effective action. The Mellin space
approach is an "on-shell method" inspired by the close analogy between
holographic correlators and flat space scattering amplitudes. We conjecture a
compact formula for the four-point correlators of one-half BPS single-trace
operators of arbitrary weights. Our general formula has the expected analytic
structure, obeys the superconformal Ward identity, satisfies the appropriate
asymptotic conditions and reproduces all the previously calculated cases. We
believe that these conditions determine it uniquely.
