There are significant financial and legal implications for ensuring design correctness and safety in autonomous systems. This book introduces new verification and synthesis algorithms to provide certifiable trusts for real-world autonomous systems. On the theoretical front, the techniques are armed with soundness, precision, and relative completeness guarantees. On the experimental side, this book shows that techniques can be successfully applied on a sequence of real-world problems, including a suite of Toyota engine control modules verified for the first time, satellite control systems, and autonomous driving and ADAS-based maneuvers. Insights throughout the book provide a level of assurance that can be provided by formal methods for today's autonomous systems. Verification and synthesis for typical models of real-world autonomous systems are challenging due to their high dimensionality, nonlinearities, and nondeterministic and hybrid nature. In addressing these challenges, several chapters present data-driven algorithmic verification via reachability analysis of complex hybrid systems as well as controller synthesis for dynamic systems under disturbance. The book includes the first algorithm for over-approximating reach sets of general nonlinear models with locally optimal tightness guarantees as well as algorithms to find correct-by-construction controllers for nonlinear dynamical systems. It is written for researchers in the corporate world, academia, government, and practitioners in autonomous systems.