Challenging science requirements and complex space missions are driving the need for fast and robust space trajectory design and simulation tools. The main aim of this thesis is to develop new and improved high performance algorithms and solution techniques for commonly encountered problems in astrodynamics. Five major problems are considered and their state-of-the art algorithms are systematically improved. Theoretical and methodological improvements are combined with modern computational techniques, resulting in increased algorithm robustness and faster runtime performance. The five selected problems are 1) Multiple revolution Lambert problem, 2) High-fidelity geopotential (gravity field) computation, 3) Ephemeris computation, 4) Fast and accurate sensitivity computation, and 5) High-fidelity multiple spacecraft simulation. The work being presented enjoys applications in a variety of fields like preliminary mission design, high-fidelity trajectory simulation, orbit estimation and numerical optimization. Other fields like space and environmental science to chemical and electrical engineering also stand to benefit.