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 nontrivial 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.