Self-driving vehicles have the potential to revolutionize our transportation infrastructure, but also raise major challenges for research and policy development. Led by the efforts of Google and many leading automobile manufacturers, interest in the field of self-driving vehicles has surged in the past several years. In this talk, I will describe some of the core technologies of mobile robotics that underpin self-driving vehicles and other new applications of robotics, including: sensing, mapping, navigation, and path-planning. Despite recent progress, fully autonomous driving remains extremely difficult, with a number of difficult open questions. I will also describe the role of Simultaneous Localization and Mapping (SLAM) algorithms in self-driving, and will discuss potential connections of self-driving and SLAM with Deep Learning algorithms and technology.