This book is the first comprehensive treatment of Painleve differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution of Painleve transcendents. The subsequent main part of the book is devoted to topics of classical background such as representations and expansions of solutions, solutions of special type like rational and special transcendental solutions, Backlund transformations and higher order analogues, treated separately for each of these six equations. The final chapter offers a short overview of applications of Painleve equations, including an introduction to their discrete counterparts. Due to the present important role of Painleve equations in physical applications, this monograph should be of interest to researchers in both mathematics and physics and to graduate students interested in mathematical physics and the theory of differential equations.