This book is the first monograph in the field of uniqueness theorv of meromorphic functions dealing with conditions under which there is the unique function satisfying given hypotheses.Developed by R.Nevanlinna,a Finnish mathematician,early in the 1920's,research in the fleld has developed rapidly over the past three decades with a great deal of fruitful results.This book systematically summarizes the most important results in the field,including many of the authors'own previously unpublished results.In addition,useful skills and simple proofs are introduced.This book is suitable for higher level and graduate students who have a basic grounding in complex analysis,but will also appeal to researchers in mathematics.