Self-driving technologies have matured and improved to the point that, in the past few years, self-driving cars have been able to safely drive an impressive number of kilometers. It should be noted though that, in all cases, the driver seat was never empty: a human driver was behind the wheel, ready to take over whenever the situation dictated it. This is an interesting paradox since the point of a self-driving car is to remove the most unreliable part of the car, namely the human driver. So, the question naturally arises: will the driver seat ever be empty? Besides legal liability issues, the answer to that question may lie in our ability to improve the self-driving technologies to the point that the human driver can safely be removed from the driving loop altogether. However, things are not that simple. Motion safety, i.e. the ability to avoid collisions, is the critical aspect concerning self-driving cars and autonomous vehicles in general. Before letting self-driving cars transport people around (and move among them) in a truly autonomous way, it is crucial to assess their ability to avoid collision, and to seek to characterize the levels of motion safety that can be achieved and the conditions under which they can be guaranteed. All these issues are explored in this article.