2014 Reprint of 1963 Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. Olga Aleksandrovna Ladyzhenskaya was a Soviet and Russian mathematician. She was known for her work on partial differential equations (especially Hilbert's 19th problem) and fluid dynamics. She provided the first rigorous proofs of the convergence of a finite difference method for the Navier-Stokes equations. This is a revised and updated edition of a book of fundamental importance in the rigorous theory of solutions of the Navier-Stokes equations. The author considers the questions of their existence and uniqueness when satisfying appropriate boundary conditions. For this purpose she extends the class of permissible functions from the infinitely differentiable class (classical solutions) to a class of generalized functions defined in the distributional sense. Thus existence of solution in the new class is a necessary but not sufficient condition for existence in the classical sense. Linear and non-linear, steady and unsteady forms of the equations and both finite and infinite domains are all considered: in each type of problem important theorems are established in the course of which many new ideas and methods are developed. The book is strongly recommended to mathematicians interested in modern analysis and the rigorous theory of fluid mechanics.