<p>Holographic correlators on the celestial sphere of Minkowski space were recently defined in [1] as the extrapolation of bulk time-ordered correlation functions to the celestial sphere. In this work we explore the Mellin representation of such celestial correlators, which is based on the Mellin representation of conformal correlators introduced by Mack in 2009. Perturbative celestial Mellin amplitudes have some similarities with Mellin amplitudes for AdS Witten diagrams: they are meromorphic functions of the Mellin variables, where contact diagrams have polynomial Mellin amplitudes and particle exchanges are encoded by a specific set of poles. We find the Mellin representation to be a useful tool to study and to compute celestial correlators, and we give various examples in scalar field theories at both tree and loop level, for both massive and massless fields. We also discuss the non-perturbative structure of celestial Mellin amplitudes following from the Källén-Lehmann spectral representation of bulk two-point functions.</p>