This book was originally published in 1977 and has since been reprinted four times (the last reprint was in 1984). The current volume is reprinted and fully retypeset by the AMS. It is very close in content to the 1984 edition. The book presents a systematic treatment of results on the theory and numerical analysis of the Navier-Stokes equations for viscous incompressible fluids. Considered are the linearized stationary case, the nonlinear stationary case, and the full nonlinear time-dependent case. The relevant mathematical tools are introduced at each stage.The new material in this book is Appendix III, reproducing a survey article written in 1998. This appendix contains a few aspects not addressed in the earlier editions, in particular a short derivation of the Navier-Stokes equations from the basic conservation principles in continuum mechanics, further historical perspectives, and indications on new developments in the area. The appendix also surveys some aspects of the related Euler equations and the compressible Navier-Stokes equations. Readers are advised to peruse this appendix before reading the core of the book. This book presents basic results on the theory of Navier-Stokes equations and, as such, continues to serve as a comprehensive reference source on the topic.