In this plenary lecture design problems where true synthesis procedures are available for linear systems will be reviewed. A classic example of this class of problems is the linear-quadratic-regulator problem. However for many practical design problems, with fixed controller structure, no true synthesis procedures are available. Many practical design problems can be reduced mathematically to the study of quantified multivariable polynomial inequalities. For this class of problems, often only probabilistic (Monte Carlo) algorithms are computable. This means that solutions to the design problem are found by making enough random guesses. Recently strong interest in this probabilistic approach to design has developed, under the name, “statistical learning theory”. This probabilistic theory, and some intermediate deterministic approaches to difficult practical problems will be presented and critiqued. In particular three approaches will be reviewed and compared, symbolic computer methods, deterministic branch-and-bound methods, and Monte Carlo methods.