This is the first part of a comprehensive two-volume treatment that reviews current knowledge concerning the kinetic theory of polyatomic gases. Recent years have seen intensive research in this area with many interesting results, however these findings are not generally known and are frequently omitted from textbooks on kinetic theory. This work responds directly to that need, and is the first to provide a comprehensive account of both theoretical and experimental aspects of the behavior of polyatomic gases. The first volume deals with dilute gases in which boundary-layer effects can be neglected. Transport and relaxation phenomena such as heat conduction, diffusion, thermal diffusion, and viscous flow are considered from the point of view of kinetic theory. Special attention is paid to phenomena which do not occur in monatomic gases, including light scattering, nuclear spin relaxation, microwave absorption, and the effects of magnetic and electric fields on transport properties. Detailed theoretical derivations are presented throughout the book, and experimental results and methods are documented in tables and figures. Introductory sections will be useful for graduate students with a knowledge of elementary classical and quantum mechanics. The more specialized chapters are aimed at professionals who want information regarding the present state of the art in a particular field or who require data for various phenomena. Volume 2, which will shortly follow the publication of this first volume, establishes the connections between molecular scattering theory and effective cross-sections in dilute gases, then considers rarefied gases in detail.