This book presents a number of techniques for robustness analysis of uncertain systems. The theoretical basis for their development is derived from the application of convex optimization tools to problems involving positivity of homogeneous polynomial forms. Usually, stability and performance analysis of dynamic systems affected by structured uncertainties, requires the solution of non-convex optimization problems. Constructing a family of convex relaxations, namely convex optimisations, can provide upper or lower bound to the original problem. In this book convex relaxations for several robustness problems are derived by exploiting and providing new results on the theory of homogeneous polynomial forms. A framework is introduced for dealing with positivity of homogeneous forms via the solution of linear matrix inequalities, with examples including Lyapunov analysis of uncertain systems, computation of the parametric robust stability margin, and robust performance analysis for polytopic systems.