This comprehensive monograph is devoted to the study of polynomially convex sets, which play a key role in the theory of functions of several complex variables. The presentation includes the general properties of polynomially convex sets with particular attention to the theory of the hulls of one-dimensional sets. The theory is motivated with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries. This beautiful exposition of a rich and complex theory, which contains much material not available in other texts and which is destined to be the standard reference for many years, will appeal to all those with an interest in multivariate complex analysis.