Conformal invariance places powerful constraints on the properties of a quantum field theory.  In position space, the form of 2- and 3-point correlation functions is completely fixed by this symmetry up to just a few constants.  In this talk, we examine the corresponding story in momentum space.  Starting from first principles, we show how to construct the momentum-space 2- and 3-point functions of a general conformal field theory.  For certain space-time and operator dimensions non-trivial renormalization is required, meaning one cannot simply Fourier transform from position space.  We show how to perform this renormalisation directly in momentum space leading to novel conformal anomalies and beta functions.  The results have potential applications to many fields including early universe cosmology, holographic dualities, and condensed matter physics.