  We study the $6j$ symbol for the conformal group, and its appearance in three
seemingly unrelated contexts: the SYK model, conformal representation theory,
and perturbative amplitudes in AdS. The contribution of the planar Feynman
diagrams to the three-point function of the bilinear singlets in SYK is shown
to be a $6j$ symbol. We generalize the computation of these and other Feynman
diagrams to $d$ dimensions. The $6j$ symbol can be viewed as the crossing
kernel for conformal partial waves, which may be computed using the Lorentzian
inversion formula. We provide closed-form expressions for $6j$ symbols in
$d=1,2,4$. In AdS, we show that the $6j$ symbol is the Lorentzian inversion of
a crossing-symmetric tree-level exchange amplitude, thus efficiently packaging
the double-trace OPE data. Finally, we consider one-loop diagrams in AdS with
internal scalars and external spinning operators, and show that the triangle
diagram is a $6j$ symbol, while one-loop $n$-gon diagrams are built out of $6j$
symbols.
