Novel universal quantities can arise in scaling terms sub-leading to the area law, in the entanglement entropy in quantum critical theories in d > 1+1. These depend crucially on the geometry of the entangled bipartition. In some cases, they are known to be related to scaling dimensions obtained from conventional n-point functions, but in some cases no clear relationships are yet known. I will survey the results that we have collected over the last several years, based partially on numerical calculations at free and interacting quantum critical points, and partially on theoretical scaling arguments for the Renyi entanglement entropies.