Discontinuous dynamical systems arise in many applications, including optimal control, robotics, and mechanics. These systems involve discontinuities in the vector field, which can occur naturally or be intentionally designed for control purposes. Discontinuous feedback is often necessary for stabilization when continuous feedback fails. Examples include sliding mode control, robotic swarms, and nonsmooth harmonic oscillators. Classical solutions, which are continuously differentiable, are not always valid for discontinuous systems. Instead, alternative notions of solutions, such as Caratheodory and Filippov solutions, are used. Caratheodory solutions are absolutely continuous and satisfy the differential equation almost everywhere, while Filippov solutions are defined by considering the set of directions of the vector field around each point. Filippov solutions are particularly useful for handling discontinuities and ensuring the existence of solutions. The existence and uniqueness of solutions depend on the properties of the vector field. For example, one-sided Lipschitz conditions can guarantee uniqueness. However, discontinuous systems may require different solution notions. Filippov solutions are widely used in applications such as sliding mode control and mechanics with Coulomb friction. The stability of discontinuous systems is analyzed using nonsmooth analysis, including generalized gradients and proximal subdifferentials. These tools help characterize the behavior of solutions and provide conditions for stability. Stability results for discontinuous systems often differ from those for smooth systems, requiring careful consideration of the solution notion. The article discusses various solution notions, including Caratheodory, Filippov, and sample-and-hold solutions, and their properties. It also addresses the existence, uniqueness, and stability of solutions for discontinuous dynamical systems, emphasizing the importance of appropriate solution definitions in analyzing such systems.Discontinuous dynamical systems arise in many applications, including optimal control, robotics, and mechanics. These systems involve discontinuities in the vector field, which can occur naturally or be intentionally designed for control purposes. Discontinuous feedback is often necessary for stabilization when continuous feedback fails. Examples include sliding mode control, robotic swarms, and nonsmooth harmonic oscillators. Classical solutions, which are continuously differentiable, are not always valid for discontinuous systems. Instead, alternative notions of solutions, such as Caratheodory and Filippov solutions, are used. Caratheodory solutions are absolutely continuous and satisfy the differential equation almost everywhere, while Filippov solutions are defined by considering the set of directions of the vector field around each point. Filippov solutions are particularly useful for handling discontinuities and ensuring the existence of solutions. The existence and uniqueness of solutions depend on the properties of the vector field. For example, one-sided Lipschitz conditions can guarantee uniqueness. However, discontinuous systems may require different solution notions. Filippov solutions are widely used in applications such as sliding mode control and mechanics with Coulomb friction. The stability of discontinuous systems is analyzed using nonsmooth analysis, including generalized gradients and proximal subdifferentials. These tools help characterize the behavior of solutions and provide conditions for stability. Stability results for discontinuous systems often differ from those for smooth systems, requiring careful consideration of the solution notion. The article discusses various solution notions, including Caratheodory, Filippov, and sample-and-hold solutions, and their properties. It also addresses the existence, uniqueness, and stability of solutions for discontinuous dynamical systems, emphasizing the importance of appropriate solution definitions in analyzing such systems.