The covariant renormalization of composite operators leads to Weyl anomalies. As a consequence, in the effective action the operators acquire an anomalous dependence on the conformal factor of the metric. The questions I will address in this talk are whether this anomalous dependence can be computed systematically and whether it can have interesting consequences for cosmology.

I will first present a two-dimensional gravitational model, where the anomaly for the cosmological constant can be computed explicitly using results from Liouville theory and leads to a non-local quantum effective action. The resulting quantum energy momentum tensor is non-local and leads to a deflating vacuum energy in a homogeneous and isotropic expanding universe. I will then turn to four dimensions, and discuss the computation of the analogous non-local effective action encoding the Weyl anomalies in four dimensions and some of its possible cosmological implications.