A new, novel numerical optimization algorithm is developed, tested, and used to solve difficult numerical problems from the field of astrodynamics. First, a brief review of optimization theory is presented and common numerical optimization techniques are discussed. Then, the new method, called the Learning Approach to Sampling Optimization (LA) is presented. Simple, illustrative examples are given to further emphasize the simplicity and accuracy of the LA method. Benchmark functions in lower dimensions are studied and the LA is compared, in terms of performance, to widely used methods. Three classes of problems from astrodynamics are then solved. First, the N - impulse orbit transfer and rendezvous problems are solved by using the LA optimization technique along with derived bounds that make the problem computationally feasible. This marriage between analytical and numerical methods allows an answer to be found for an order of magnitude greater number of impulses than are currently published. Next, the N -impulse work is applied to design periodic close encounters (PCE) in space. The encounters are defined as an open rendezvous, meaning that two spacecraft must be at the same position at the same time, but their velocities are not necessarily equal. The PCE work is extended to include N -impulses and other constraints, and new examples are given. Finally, a trajectory optimization problem is solved using the LA algorithm and comparing performance with other methods based on two models-with varying complexity-of the Cassini-Huygens mission to Saturn. The results show that the LA consistently outperforms commonly used numerical optimization algorithms.