This volume commemorates the work of Alan Turing, who not only introduced the most influential concept of a machine model of effective computability, but who also anticipated in his work the diversity of topics brought together here. Among his major contributions, Turing's "On Computable Numbers, With an Application to the Entscheidungsproblem," first published in 1937, is acknowledged as a landmark of the computer age. Part I of this volume explores historical aspects with essays on background, on Turing's work, and on subsequent developments. Part II contains an extensive series of essays on the influence and applications of these ideas in mathematics, mathematical logic, philosophy of mathematics, computer science, artificial intelligence, philosophy of language, philosophy of mind, and physics.