The effective action of gravity is supposed to contain the main information about quantum corrections to gravity. However, in many cases it cannot be calculated exactly. A remarkable and important exception is the effective action of vacuum for massless and conformal-invariant matter fields. In this case the effective action can be easily derived by integrating the trace anomaly. The integration constant is an unknown conformal functional of the background metric, but for zero-order cosmology this functional is irrelevant and the solution becomes exact. Until recently the integrated anomaly was known only in dimensions d=2 and d=4, but recently we achieved the explicit result for d=6, which confirms the universal functional
structure of the effective action.