All chapters are united by a common concept: together with the reader, go from illustrative examples of number-theoretic objects and problems, through general concepts and theories that have developed over time, to some of the latest achievements and visions of modern mathematics and sketches for further research. New sections written for this edition include a condensed exposition of Wiles' proof of Fermat's large theorem, a recently discovered polynomial algorithm for checking primality, a review of the count of rational points on manifolds, and other subjects; the final part of the book is devoted to arithmetic cohomology and non-commutative geometry.