This book teaches the art of enumeration, or counting, by leading the reader through a series of carefully chosen problems that are arranged strategically to introduce concepts in a logical order and in a provocative way. It is organized in eight sections, the first four of which cover the basic combinatorial entities of strings, combinations, distributions, and partitions. The last four cover the special counting methods of inclusion and exclusion, recurrence relations, generating functions, and the methods of Pуlya and Redfield that can be characterized as "counting modulo symmetry." The unique format combines features of a traditional textbook with those of a problem book. The subject matter is presented through a series of approximately 250 problems, with connecting text where appropriate, and is supplemented by approximately 200 additional problems for homework assignments. Many applications to probability are included throughout the book. While intended primarily for use as the text for a college-level course taken by mathematics, computer science, and engineering students, the book is suitable as well for a general education course at a good liberal arts college, or for self study.