This book contains papers presented at the AMS-IMS-SIAM Joint Summer Research Conference on Current Trends in Arithmetical Algebraic Geometry, held in August 1985 at Humboldt State University in Arcata, California. The conference focused on hyperbolic geometry, Arak�lov theory, and connections between �tale cohomology and crystalline cohomology. The book is accessible to both graduate students and mathematicians interested in current topics in arithmetical geometry, particularly those readers in neighboring fields who wish to acquire an overview of some topics in which research is now intensely active. Some of the introductory papers will be of interest to the nonspecialists, while others are directed at researchers and advanced graduate students familiar with the area. Portions of this book are likely to become fundamental references and will be of permanent value to researchers..