The book consists of seven chapters: In Chapter 1, we introduce some basic notation and terminology on fields and algebraic geometry which are mainly used to explain clearly the topics in Chapter 3 related to number theory. Chapter 2 is a foundation of value distribution theory which is used in Chapter 4, Chapter 6 and Chapter 7 to introduce the analogues related to number theory in Nevanlinna theory, say, abc-problems, meromorphic solutions of Fermat’s equations and the Waring problem, Green-Griffiths’ conjecture, Griffiths’ and Lang’s conjectures, Riemann’s ζ-function, and so on. Chapter 5 contains value distribution theory over non-Archimedean fields and some applications related to topics in number theory. Moreover, a few equidistribution formulae illustrating the differences with the classical Nevanlinna theory have been exhibited. Each chapter of this book is self-contained and this book is appended with a comprehensive and up-dated list of references. The book will provide not just some new research results and directions but challenging open problems in studying Diophantine approximation and Nevanlinna theory. One of the aims of this book is to make timely surveys on these new results and their related developments; some of which are newly obtained by the authors and have not been published yet. It is hoped that the publication of this book will stimulate, among our peers, further researches on Nevanlinna’s value distribution theory, Diophantine approximation and their applications.