  We propose and demonstrate a new use for conformal field theory (CFT)
crossing equations in the context of AdS/CFT: the computation of loop
amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in
AdS are largely unexplored, mostly due to technical difficulties in direct
calculations. We revisit this problem, and the dual $1/N$ expansion of CFTs, in
two independent ways. The first is to show how to explicitly solve the crossing
equations to the first subleading order in $1/N^2$, given a leading order
solution. This is done as a systematic expansion in inverse powers of the spin,
to all orders. These expansions can be resummed, leading to the CFT data for
finite values of the spin. Our second approach involves Mellin space. We show
how the polar part of the four-point, loop-level Mellin amplitudes can be fully
reconstructed from the leading-order data. The anomalous dimensions computed
with both methods agree. In the case of $\phi^4$ theory in AdS, our crossing
solution reproduces a previous computation of the one-loop bubble diagram. We
can go further, deriving part of the four-point function in $\phi^3+\phi^4$
theory in AdS which had never been computed. In the process, we show how to
analytically derive anomalous dimensions from Mellin amplitudes with an
infinite series of poles, and discuss applications to more complicated cases
such as the ${\cal N}=4$ super-Yang-Mills theory.
