The Purpose of this volume is to provide an account of the modern algebraic methods available for the investigation of the birational geometry of algebraic varieties. An account of these methods has already been published by Professor Andre Weil in his Foundations of Algebraic Geometry （New York， 1946）， and when Professor Zariski's Colloquium Lectures， delivered in 1947 to the American Mathematical Society， are published， another full account of this branch of geometry will be available. The excuse for a third work dealing with this subject is that the present volume is designed to appeal to a different class of reader. It is written to meet the needs of those geometers trained in the classical methods of algebraic geometry who are anxious to acquire the new and powerful tools provided by modern algebra， and who also want to see what they mean in terms of ideas familiar to them. Thus in this volume we are primarily concerned with methods， and not with the statement of original results or with a unified theory of varieties.Such a purpose in writing this volume has had several effects on the plan of the work. In the first place， we have confined our attention to varieties defined over a ground field without characteristic. This is partly because the geometrical significance of the algebraic methods and results is more easily comprehended by a classical geometer in this case; also， though others have shown that modern algebraic methods have enabled us to make great strides in the theory of algebraic varieties over a field of finite characteristic， many of the theorems which the classical geometer regards as fundamental have only been proved， as yet， in the restricted case.