A variety of optimal trajectories from a circular low-Earth parking orbit to a circular low-lunar parking orbit are computed for a range of low-thrust spacecraft. The problem is studied in the context of the classical restricted three-body problem. Minimum-fuel, planar trajectories with a fixed thrust-coast-thrust engine sequence are computed for both a "high-end" low-thrust spacecraft and "moderate" low-thrust nuclear electric propulsion (NEP) spacecraft. Since a low-thrust trajectory is a long duration transfer with slowly developing spirals about the Earth and Moon, the minimum-fuel Earth-Moon trajectory is obtained by formulating and successively solving a hierarchy of sub-problems. The subproblems include optimal Earth-escape and Moon-capture trajectories and sub-optimal translunar trajectories. The complete minimum-fuel trajectory problem is eventually solved using a "hybrid" direct/indirect method which utilizes the benefits of a direct optimization method and an indirect method from optimal control theory. Minimum-fuel transfers are also computed using a switching function structure which results in multiple thrust and coast arcs. In addition, a new combined vehicle and trajectory optimization problem of maximum payload fraction is formulated and solved. Finally, three-dimensional minimum-fuel trajectories are obtained for both the "high-end" and "moderate" low-thrust spacecraft. Numerical results are presented for various optimal Earth-Moon trajectories.