Summary: We show that the simplest FLRW cosmological system consisting in the homogeneous and isotropic massless Einstein-scalar system enjoys a hidden conformal symmetry under the 1D conformal group \(\mathrm{SL} (2, \mathbb{R})\) acting as Möbius transformations in proper time. This invariance is made explicit through the mapping of FLRW cosmology onto conformal mechanics. On the one hand, we identify the corresponding conformal Noether charges, as combinations of the Hamiltonian scalar constraint, the extrinsic curvature and the 3D volume, which form a closed \(\mathfrak{s} \mathfrak{l} (2, \mathbb{R})\) Lie algebra. On the other hand, this approach allows to write FLRW cosmology in terms of a \(\mathrm{AdS}_2\) phase space and a Schwarzian action. Preserving this conformal structure at the quantum level fixes the ordering ambiguities in the Wheeler-de Witt quantization and allows to formulate FLRW quantum cosmology as a \(\mathrm{CFT}_1\). We show that the CFT two-points correlator is realized as the overlap of the evolution in proper time of cosmological coherent wave-packets. In particular, the two-points function is built from a vacuum state which, although not conformally invariant, coincides with the cosmological vacuum annihilated by the scalar constraint. These results suggest new perspectives in classical and quantum cosmology, among which the possibility to apply the conformal bootstrap program to quantize cosmological backgrounds.