  We present a Feynman integral representation for the general momentum-space
scalar $n$-point function in any conformal field theory. This representation
solves the conformal Ward identities and features an arbitrary function of
$n(n-3)/2$ variables which play the role of momentum-space conformal
cross-ratios. It involves $(n-1)(n-2)/2$ integrations over momenta, with the
momenta running over the edges of an $(n-1)$-simplex. We provide the details in
the simplest non-trivial case (4-point functions), and for this case we
identify values of the operator and spacetime dimensions for which
singularities arise leading to anomalies and beta functions, and discuss
several illustrative examples from perturbative quantum field theory and
holography.
