The volume presents a collection of papers written in honour of Peter Swinnerton-Dyer on the occasion of his 75 birthday. The collection contains 14 papers discussing contributions of this versatile personality in number theory and algebraic geometry and also his “political commitments”. The papers are written by leading personalities in the subject. Here, the reader can find a survey of material on delicately intertwined aspects of analytic, geometric and cohomological methods, from the point of view of their role in the solution of fundamental questions of the theory of Diophantine equations (e.g., rational points on algebraic varieties, Shafarevich-Tate groups, Zagier conjecture, Diophantine approximation, properties of Abelian and Fano varieties, etc.). The papers contain a description of a variety of important results and techniques and will provide invaluable help not only for specialists in the field but also for those who would like to find a quick and competent answer to specific questions or an orientation in this important part of contemporary mathematics.