This book deals with the solutions of a group of questions related both to the general theory of transcendental numbers and to the metrical theory of diophantine (and also algebraic) approximations. The fundamental problem in this field has been known in the literature since 1932 as Mahler's conjecture. The main result of this book is a proof of Mahler's conjecture and some analogous theorems.

In Part I, the "Classical" case of Mahler's conjecture, dealing with real and complex numbers, is considered. This part should be comprehensible to any who knows the elements of measure theory and possesses sufficient perseverance in over-coming purely logical difficulties. Part II is concerned with locally compact fields with nonarchimedean valuation. This part requires a general familiarity with the structure of fields with nonarchimedean valuation. All the necessary information is given in the text with references to the sources.