The book starts with an elementary introduction to formal languages appealing to the intuition of working mathematicians and unencumbered by philosophical or normative prejudices such as those of constructivism or intuitionism. It proceeds to the Proof Theory and presents several highlights of Mathematical Logic of 20th century: Gödel's and Tarski's Theorems, Cohen's Theorem on the independence of Continuum Hypothesis. Unusual for books on logic is a section dedicated to quantum logic. Then the exposition moves to the Computability Theory, based on the notion of recursive functions and stressing number{theoretic connections. A complete proof of Davis{Putnam{Robinson{Matiyasevich theorem is given, as well as a proof of Higman's theorem on recursive groups. Kolmogorov complexity is treated. The third Part of the book establishes essential equivalence of proof theory and computation theory and gives applications such as Gödel's theorem on the length of proofs. The new Chapter IX, written for the second edition, treats, among other things, categorical approach to the theory of computation, quantum computation, and P/NP problem. The new Chapter X, written for the second edition by Boris Zilber, contains basic results of Model Theory and its applications to mainstream mathematics. This theory found deep applications in algebraic and Diophantine geometry. Yuri Ivanovich Manin is Professor Emeritus at Max-Planck-Institute for Mathematics in Bonn, Germany, Board of Trustees Professor at the Northwestern University, Evanston, USA, and Principal Researcher at the Steklov Institute of Mathematics, Moscow, Russia. Boris Zilber, Professor of Mathematics at the University of Oxford, has been added to the second edition.