Unconventional high-Tc superconductivity, defined both in terms of the magnitude of the superconducting transition temperature, Tc, and the key role played by electronic correlations, not only is the realm of atom-based low-dimensional layered systems such as the cuprates or the iron pnictides but is also accessible in molecular systems such as the cubic alkali fullerides with stoichiometry A3C60 (A=alkali metal). In fulleride superconductors, isotropic high-Tc superconductivity occurs in competition with electronic ground states resulting from a fine balance between electron correlations and electron-phonon coupling in an electronic phase diagram strikingly similar to those of unconventional superconductors such as the cuprates and the heavy fermions. Superconductivity at the highest Tc (38 K) known for any molecular material emerges from the antiferromagnetic insulating state solely by changing an electronic parameter - the overlap between the outer wave functions of the constituent molecules - and Tc scales universally in a structure-independent dome-like relationship with proximity to the Mott metal-insulator transition (quantified by V, the volume/C60, or equivalently by (U/W), the ratio of the on-site Coulomb energy, U, to the electronic bandwidth, W), a hallmark of electron correlations characteristic of high-Tc superconductors other than fullerides. The C60 molecular electronic structure plays a key role in the Mott-Jahn-Teller (MJT) insulator formed at large V, with the on-molecule dynamic Jahn-Teller (JT) effect distorting the C60 3- units and quenching the t1u orbital degeneracy responsible for metallicity. As V decreases, the MJT insulator transforms first into an unconventional correlated JT metal (where localised electrons coexist with metallicity and the on-molecule distortion persists) and then into a Fermi liquid with a less prominent molecular electronic signature. This normal state crossover is mirrored in the evolution of thesuperconducting state, with the highest Tc found at the boundary between unconventional correlated and conventional weak-coupling BCS superconductivity, where the interplay between extended and molecular aspects of the electronic structure is optimised to create the superconductivity dome.