In the three years since the Russian edition of this book was written, quite a few papers devoted to a mathematically rigorous analysis of nonstationary solutions of the Navier-Stokes equations have been published. These papers either pursue the investigation of differential properties of the solutions whose existence and uniqueness is proved in the present book, or else they give other methods for obtaining such solutions. However, the basic problem of the unique solvability "in the large" of the boundary-value problem for the general three-dimensional nonstationary Navier-Stokes equations (with no assumptions other than a certain smoothness of the initial field and of the external forces) remains as open as ever.