In 2014, the first meeting of the Verification of Autonomous Systems Working Group was held in the naïve hope that if we assembled roboticists, computer scientists, electrical and mechanical engineers, mathematicians, and verification specialists (along with at least one translator), we could explain our verification problem to the mathematicians and either they would identify a mathematical space or representation that would help us handle our complexity problem, or they would determine that it is theoretically impossible for a mathematical space or representation to help handle our complexity problem, or they would determine that further work was necessary to figure out whether a mathematical representation that could help manage our complexity was possible. Instead, what actually came out of the meeting was a list of 27 open research challenges, grouped into four main categories. Over the next few years, we met with one another, shared funding opportunities, identified new challenges, and made progress in addressing them. We established an IEEE Robotics and Automation Society Technical Committee and we’re developing a guide to best practices under the auspices of the IEEE Standards Association. This book captures the state of research into these challenges as the initial research began to bear fruit.