In 2019 we celebrated the 50th anniversary of the historical Apollo 11 lunar landing. By striving to return human to the Moon by 2024, NASA's Artemis program aims at demonstrating new technologies and capabilities needed for future exploration including Mars. The Apollo program gave the birth to the technical area of aerospace guidance as we know it today. In this presentation we will examine the role of onboard computation in aerospace guidance technology from the Apollo era to today. Fueled by the tremendous increase of onboard computational power, we are witnessing transformational changes in guidance and control technology where traditional designs are replaced by a new paradigm known as Computational Guidance and Control (CG&C). Different from other computational branches of engineering and sciences, the computation in CG&C must take place onboard, mostly in real time. CG&C allows much more complex guidance and control tasks to be performed than ever before, offering great potential for significant increase in autonomy, capability, and performance without the need for additional hardware or increase in system complexity. This presentation will focus on the transformation of the guidance methods and algorithms for space transportation systems over the last 5 decades, driven by the demand for higher performance and enabled by advances in theory and computational power. Examples of legacy technology and state-of-the-art guidance algorithms will be provided and contrasted, and applications in human-scale Mars missions will be demonstrated.