Magnetic molecules are prototypical systems to investigate peculiar quantum mechanical phenomena. As such, simulating their static and dynamical behavior is intrinsically difficult for a classical computer, due to the exponential increase of required resources with the system size. Quantum computers solve this issue by providing an inherently quantum platform, suited to describe these magnetic systems. Here, we show that both the ground state properties and the spin dynamics of magnetic molecules can be simulated on prototype quantum computers, based on superconducting qubits. In particular, we study small-size anti-ferromagnetic spin chains and rings, which are ideal test-beds for these pioneering devices. We use the variational quantum eigensolver algorithm to determine the ground state wave-function with targeted ansatzes fulfilling the spin symmetries of the investigated models. The coherent spin dynamics are simulated by computing dynamical correlation functions, an essential ingredient to extract many experimentally accessible properties, such as the inelastic neutron cross-section.