This book presents the mathematical theory of turbulence to engineers and physicists, and the physical theory of turbulence to mathematicians. It is the result of many years of research by the authors to analyze turbulence using Sobolev spaces and functional analysis. In this way the authors have recovered parts of the conventional theory of turbulence, deriving rigorously from the Navier-Stokes equations that had been arrived at earlier by phenomenological arguments. Appendices give full details of the mathematical proofs and subtleties.