Despite their importance and the amount of research published on the t heory of the Navier-Stokes equations, some very elementary and fundame ntal questions remain unanswered. In this book, the author explores tw o main topics: the search for mild solutions using the tools of real h armonic analysis, and the 1982 regularity criterion of Caffanelli et a l., which has produced many results on the existence or non-existence of solutions. The book includes a self-contained introduction to real harmonic analysis, addresses both strong solutions by direct methods a nd weak solutions by indirect...