This paper introduces a new probabilistic framework for robust control analysis and synthesis, focusing on problems that can be formulated as minimizing a linear objective subject to convex constraints parameterized by uncertain terms. The approach, called the "scenario approach," involves sampling a finite number of uncertainty instances (scenarios) and solving a standard convex optimization problem (the scenario problem) for each. The solution to the scenario problem is shown to be approximately feasible for the original robust problem with high probability, and the number of required samples is efficiently bounded. This method provides a viable and implementable way to achieve desired levels of robustness in control design, addressing the computational hardness and conservatism issues of traditional worst-case approaches. The paper also discusses specific control problems, such as generalized quadratic stability and state-feedback stabilization, where the scenario approach can be applied to solve them efficiently.This paper introduces a new probabilistic framework for robust control analysis and synthesis, focusing on problems that can be formulated as minimizing a linear objective subject to convex constraints parameterized by uncertain terms. The approach, called the "scenario approach," involves sampling a finite number of uncertainty instances (scenarios) and solving a standard convex optimization problem (the scenario problem) for each. The solution to the scenario problem is shown to be approximately feasible for the original robust problem with high probability, and the number of required samples is efficiently bounded. This method provides a viable and implementable way to achieve desired levels of robustness in control design, addressing the computational hardness and conservatism issues of traditional worst-case approaches. The paper also discusses specific control problems, such as generalized quadratic stability and state-feedback stabilization, where the scenario approach can be applied to solve them efficiently.