This thesis research examines the fuel optimal trajectories for a spacecraft going to Mars and returning to Earth. Challenges encountered include the defining of equations of motion and optimality conditions, formulation of constraints, and overcoming difficulties deriving from large distances and times involved and the accuracy required. In addition to calculating an optimal trajectory, two different arrival orbits at Mars are compared: a clockwise and counterclockwise Martian orbit. The optimal trajectories are computed using a mathematical optimization algorithm SNOPT, developed at Stanford University and UC San Diego. A solution method is recommended where an initial guess is generated analytically via a patched conic approach. The problem is solved in two steps: first with relaxed constraints, then using that result to find the optimal solution. This approach is proven by separately solving two different trajectories: an Earth to Mars trajectory and a Mars to Earth trajectory. The results demonstrate that the two arrival conditions are very similar in most aspects, including planetary phase angles and total trajectory time. The trajectory using the counterclockwise Mars orbit has a slight advantage in propellant usage (DeltaV), but the difference is less than 1%. These transfers also have symmetries between the outbound and return portions of the trajectories. Both trajectories should be available for consideration for a mission to Mars depending on other mission requirements.