Abstract: The robotic exploration of asteroids is an important human endeavor motivated by significant themes: the scientific study of solar system formation and evolution, the exploration of the solar system, and the protection of society against future hazardous asteroid impactors. This endeavor involves significant challenges across a range of technical issues associated with fundamental dynamics and control: development of new models for mathematically describing the asteroid environment, developing new understanding of the basic mechanics of asteroids, describing and predicting orbital motion in these highly perturbed environments, and developing novel methods of guidance and control for asteroid orbiters. This talk will introduce the technical challenges the asteroid environment poses, review progress which has occurred over the last two decades, and indicate where research is still progressing. It will also give a preview of upcoming asteroid exploration missions and their scientific and technical significance.