A central topic in cosmology, both theoretically and experimentally, is the study of the statistical properties of primordial perturbations, likely produced by inflation. Primordial perturbations are (approximately) scale-invariant. Dilations are usually related with the full conformal group SO(4,1): what is the role of these ransformations? I will show that in all models in which the source of perturbations is decoupled from the inflaton, correlation functions enjoy full conformal symmetry. Moreover, in any single-field inflation model the conformal group is non-linearly realized and this implies the existence of new conformal consistency relations.