In this book the authors reduce a wide variety of problems arising in system and control theory to a handful of convex and quasiconvex optimisation problems that involve linear matrix inequalities. These optimisation problems can be solved using recently developed numerical algorithms that not only are polynomial-time but also work very well in practice; the reduction therefore can be considered a solution to the original problems. This book opens up an important new research area in which convex optimisation is combined with system and control theory, resulting in the solution of a large number of previously unsolved problems.