Lithium-ion batteries have the potential to revolutionize the transportation industry, as they did for wireless communication. A judicious choice of the liquid electrolytes used in these systems is required to achieve a good balance among high-energy storage, long cycle life and stability, and fast charging. Ethylene-carbonate (EC) and propylene-carbonate (PC) are popular electrolytes. However, to date, almost all molecular-dynamics simulations of these fluids rely on classical force fields, while a complete description of the functionality of Li-ion batteries will eventually require quantum mechanics. We perform accurate ab initio molecular-dynamics simulations of ethylene- and propylene-carbonate with LiPF(6) at experimental concentrations to build solvation models which explain available neutron scattering and nuclear magnetic resonance (NMR) results and to compute Li-ion solvation energies and diffusion constants. Our results suggest some similarities between the two liquids as well as some important differences. Simulations also provide useful insights into formation of solid-electrolyte interphases in the presence of electrodes in conventional Li-ion batteries.