Fuel-optimal maneuvers of a constant-specific-impulse, thrust-limited spacecraft in field-free space are analyzed. The simple problem of an optimal maneuver from a state of rest at one location to a state of rest at another location becomes very complex when a path constraint is introduced. Solutions are obtained using a direct numerical optimization method that combines Hermite-Simpson transcription and non-linear programming. The resulting Lagrange multipliers provide a discrete approximation to the primer vector. The necessary conditions for an optimal solution can then be checked to validate the solution. Fundamental concepts, such as the existence of boundary arcs or boundary points and the optimal number of coast arcs, are examined. A comprehensive solution is obtained for symmetric rest-to-rest maneuvers. More general maneuvers are also examined.