Discontinuous dynamical systems are prevalent in various applications, including optimal control, robotics, and mechanics. These systems involve discontinuities in the vector field, which can arise naturally or be intentionally designed to achieve specific behaviors. The article explores the theory and analysis of such systems, focusing on solutions, nonsmooth analysis, and stability. It discusses the challenges in defining solutions for discontinuous systems and introduces different notions of solutions, such as Caratheodory and Filippov solutions. The article also addresses the existence and uniqueness of solutions, stability properties, and the role of nonsmooth analysis in understanding the behavior of discontinuous systems. It highlights the importance of appropriate solution notions, such as Filippov solutions, in analyzing systems with discontinuities, and provides examples to illustrate these concepts. The discussion includes the impact of discontinuities on system behavior, the necessity of considering different solution types, and the application of these concepts to real-world problems in control and mechanics. The article concludes with a review of key results and the importance of understanding discontinuous dynamical systems in various applications.Discontinuous dynamical systems are prevalent in various applications, including optimal control, robotics, and mechanics. These systems involve discontinuities in the vector field, which can arise naturally or be intentionally designed to achieve specific behaviors. The article explores the theory and analysis of such systems, focusing on solutions, nonsmooth analysis, and stability. It discusses the challenges in defining solutions for discontinuous systems and introduces different notions of solutions, such as Caratheodory and Filippov solutions. The article also addresses the existence and uniqueness of solutions, stability properties, and the role of nonsmooth analysis in understanding the behavior of discontinuous systems. It highlights the importance of appropriate solution notions, such as Filippov solutions, in analyzing systems with discontinuities, and provides examples to illustrate these concepts. The discussion includes the impact of discontinuities on system behavior, the necessity of considering different solution types, and the application of these concepts to real-world problems in control and mechanics. The article concludes with a review of key results and the importance of understanding discontinuous dynamical systems in various applications.