The paper proposes a probabilistic framework for robust control design, addressing problems that can be expressed as minimizing a linear objective subject to convex constraints parameterized by uncertainty. This includes NP-hard control problems represented by parameter-dependent linear matrix inequalities (LMIs). The key idea is to sample constraints to form a convex optimization problem (the scenario problem), whose solution is approximately feasible for the original infinite set of constraints. The paper provides an explicit bound on the number of samples needed to achieve a desired probabilistic guarantee of robustness. This approach allows for polynomial-time solutions to control problems that are generally hard to solve deterministically. The scenario approach is shown to be effective for robust control design, offering a probabilistic guarantee of robustness rather than a deterministic one. The paper also discusses the application of this approach to robust control problems, including analysis and synthesis via parameter-dependent Lyapunov functions and state-feedback stabilization. The scenario approach is demonstrated to be computationally efficient and effective in providing approximate solutions with high probability of feasibility.The paper proposes a probabilistic framework for robust control design, addressing problems that can be expressed as minimizing a linear objective subject to convex constraints parameterized by uncertainty. This includes NP-hard control problems represented by parameter-dependent linear matrix inequalities (LMIs). The key idea is to sample constraints to form a convex optimization problem (the scenario problem), whose solution is approximately feasible for the original infinite set of constraints. The paper provides an explicit bound on the number of samples needed to achieve a desired probabilistic guarantee of robustness. This approach allows for polynomial-time solutions to control problems that are generally hard to solve deterministically. The scenario approach is shown to be effective for robust control design, offering a probabilistic guarantee of robustness rather than a deterministic one. The paper also discusses the application of this approach to robust control problems, including analysis and synthesis via parameter-dependent Lyapunov functions and state-feedback stabilization. The scenario approach is demonstrated to be computationally efficient and effective in providing approximate solutions with high probability of feasibility.