This paper introduces control barrier functions (CBFs) and their application in verifying and enforcing safety properties in safety-critical controllers, particularly in the context of optimization-based control. The authors provide a comprehensive overview of the main technical results and discuss applications in various domains, including robotic systems. The paper begins by defining safety and its relationship with liveness, highlighting the historical development of barrier functions and their role in ensuring safety. It then delves into the theory of CBFs, explaining how they can be used to enforce safety in a minimally invasive manner. The authors present the foundations of CBFs, including their motivation from control Lyapunov functions (CLFs) and the conditions under which a function can be a CBF. They also discuss the synthesis of optimization-based controllers using CBFs and the unification of safety and stability through optimization-based methods. The paper further explores the application of CBFs in systems with actuation constraints and systems with higher relative degree, introducing exponential CBFs for enforcing safety constraints of arbitrarily high relative degree. Finally, the authors provide practical examples of CBFs in robotic systems, such as dynamic walking on stepping stones, automotive systems (adaptive cruise control and lane keeping), and dynamic balancing on Segways, demonstrating the effectiveness of CBFs in ensuring safety and stability in these applications.This paper introduces control barrier functions (CBFs) and their application in verifying and enforcing safety properties in safety-critical controllers, particularly in the context of optimization-based control. The authors provide a comprehensive overview of the main technical results and discuss applications in various domains, including robotic systems. The paper begins by defining safety and its relationship with liveness, highlighting the historical development of barrier functions and their role in ensuring safety. It then delves into the theory of CBFs, explaining how they can be used to enforce safety in a minimally invasive manner. The authors present the foundations of CBFs, including their motivation from control Lyapunov functions (CLFs) and the conditions under which a function can be a CBF. They also discuss the synthesis of optimization-based controllers using CBFs and the unification of safety and stability through optimization-based methods. The paper further explores the application of CBFs in systems with actuation constraints and systems with higher relative degree, introducing exponential CBFs for enforcing safety constraints of arbitrarily high relative degree. Finally, the authors provide practical examples of CBFs in robotic systems, such as dynamic walking on stepping stones, automotive systems (adaptive cruise control and lane keeping), and dynamic balancing on Segways, demonstrating the effectiveness of CBFs in ensuring safety and stability in these applications.