Abstract: In this paper we use Differential Evolution (DE), with best evolved results refined using a Nelder-Mead optimization, to solve complex problems in orbital mechanics relevant to low Earth orbits (LEO) and within the Earth-Moon system. A class of Lambert problems is examined to evaluate the performance and robustness of this evolutionary approach to orbit optimization. We evolve impulsive initial velocity vectors giving rise to intercept trajectories that take a spacecraft from given initial positions to specified target positions. We seek to minimize final positional error subject to time-of-flight and/or energy (fuel) constraints. We first validate that the method can recover known analytical solutions obtainable with the assumption of Keplerian motion. We then apply the method to more complex and realistic non-Keplerian problems incorporating trajectory perturbations arising in LEO due to the Earth's oblateness and rarefied atmospheric drag. Finally, a rendezvous trajectory from LEO to the L4 Lagrange point is computed. The viable trajectories obtained for these challenging problems suggest the robustness of our computational approach for real-world orbital trajectory design in LEO situations where no analytical solution exists.