The book presents the mathematical tools used to deal with problems related to slow rarefied flows, with particular attention to basic concepts and problems which arise in the study of micro- and nanomachines. The mathematical theory of slow flows is presented in a practically complete fashion and provides a rigorous justification for the use of the linearized Boltzmann equation, which avoids costly simulations based on Monte Carlo methods. The book surveys the theorems on validity and existence, with particular concern for flows close to equilibria, and discusses recent applications of rarefied lubrication theory to micro-electro-mechanical systems (MEMS). It gives a general acquaintance of modern developments of rarefied gas dynamics in various regimes with particular attention to low speed microscale gas dynamics. Senior students and graduates in applied mathematics, aerospace engineering, and mechanical mathematical physics will be provided with a basis for the study of molecular gas dynamics. The book will also be useful for scientific and technical researchers engaged in the research on gas flow in MEMS.