Control Barrier Functions (CBFs) are introduced as a tool for ensuring safety in control systems, analogous to Lyapunov functions for stability. This paper provides an overview of CBFs, their theoretical foundations, and applications in safety-critical control. The concept of CBFs is rooted in the idea of invariance of a safe set, where a function h(x) defines the safe region, and the control input u is chosen to ensure that the system remains within this region. CBFs are defined by conditions that ensure the safe set remains invariant under the system dynamics, even in the presence of disturbances or uncertainties. The paper discusses the historical development of barrier functions, starting from Nagumo's theorem in the 1940s, through the use of barrier certificates in the 1970s, and the extension to control systems in the 2000s. It highlights the importance of CBFs in addressing safety in autonomous systems, where safety is a critical design consideration. The paper also explores the integration of CBFs with control Lyapunov functions (CLFs) to achieve both stability and safety in control systems. Theoretical foundations of CBFs are presented, including their definition, properties, and synthesis of optimization-based controllers. The paper discusses the application of CBFs to systems with actuation constraints, where the safe set is defined by a function h(x), and the control input is chosen to ensure the system remains within this set. The paper also addresses the extension of CBFs to higher relative-degree safety constraints, introducing exponential CBFs that can handle arbitrarily high relative-degree safety constraints. Applications of CBFs are discussed in various domains, including robotic systems, automotive systems, and multi-agent systems. In robotic systems, CBFs are used to ensure safe walking on stepping stones, dynamic balancing on Segways, and precise foot placement. In automotive systems, CBFs are used for adaptive cruise control and lane keeping, ensuring safe following distances and maintaining the vehicle within the lane. The paper also discusses the use of CBFs in multi-agent systems, where safety constraints are enforced across multiple agents. The paper concludes with the development of optimization-based controllers that synthesize CBFs and CLFs to achieve both safety and stability in control systems. These controllers are formulated as quadratic programs, allowing for the enforcement of safety constraints while achieving desired control objectives. The results demonstrate the effectiveness of CBFs in ensuring safety in complex systems, making them a valuable tool in the design of safety-critical control systems.Control Barrier Functions (CBFs) are introduced as a tool for ensuring safety in control systems, analogous to Lyapunov functions for stability. This paper provides an overview of CBFs, their theoretical foundations, and applications in safety-critical control. The concept of CBFs is rooted in the idea of invariance of a safe set, where a function h(x) defines the safe region, and the control input u is chosen to ensure that the system remains within this region. CBFs are defined by conditions that ensure the safe set remains invariant under the system dynamics, even in the presence of disturbances or uncertainties. The paper discusses the historical development of barrier functions, starting from Nagumo's theorem in the 1940s, through the use of barrier certificates in the 1970s, and the extension to control systems in the 2000s. It highlights the importance of CBFs in addressing safety in autonomous systems, where safety is a critical design consideration. The paper also explores the integration of CBFs with control Lyapunov functions (CLFs) to achieve both stability and safety in control systems. Theoretical foundations of CBFs are presented, including their definition, properties, and synthesis of optimization-based controllers. The paper discusses the application of CBFs to systems with actuation constraints, where the safe set is defined by a function h(x), and the control input is chosen to ensure the system remains within this set. The paper also addresses the extension of CBFs to higher relative-degree safety constraints, introducing exponential CBFs that can handle arbitrarily high relative-degree safety constraints. Applications of CBFs are discussed in various domains, including robotic systems, automotive systems, and multi-agent systems. In robotic systems, CBFs are used to ensure safe walking on stepping stones, dynamic balancing on Segways, and precise foot placement. In automotive systems, CBFs are used for adaptive cruise control and lane keeping, ensuring safe following distances and maintaining the vehicle within the lane. The paper also discusses the use of CBFs in multi-agent systems, where safety constraints are enforced across multiple agents. The paper concludes with the development of optimization-based controllers that synthesize CBFs and CLFs to achieve both safety and stability in control systems. These controllers are formulated as quadratic programs, allowing for the enforcement of safety constraints while achieving desired control objectives. The results demonstrate the effectiveness of CBFs in ensuring safety in complex systems, making them a valuable tool in the design of safety-critical control systems.