This book deals primarily with the sample function behaviour of Gaussian, and related, random fields; i.e. stochastic processes whose arguments vary in a continuous fashion over some subset of N-dimensional Euclidean space. The problems that arise in describing this behaviour in the multiparameter setting are qualitatively different to those covered by the one-dimensional theory. Indeed, most of the really interesting problems are of a geometrical nature, and actually disappear in the simple case. The purpose of this book is to collect within one cover most of the contents of the substantial literature devoted to the sample function analysis of random fields, including a reasonably full and self-contained account of the geometry needed for its understanding.