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The study of uncertain systems has played a significant role throughout the history of control engineering due to unknown quantities often being present in the mathematical model of a plant. In this monograph the author provides a unified framework for the fundamental and challengingarea of robustness analysis of uncertain systems, where even the most basic problem of establishing robust stability may be still present. This framework uses linear matrix inequalities (LMIs) to exploit polynomials that can be expressed as sums of squares of polynomials (SOS). The author guides the reader through the motivations for using the framework including considering various types of uncertainties; providing guarantees for robust stability and robust performance; requiring the solution of convex optimization problems; allowing for trade-off between conservatism and complexity; and concluding with a number of special case methods. This monograph can be used by researchers and students to understand the issues and use the numerical examples to identify the use of the framework in modern controls systems.