This textbook on category theory is intended for researchers and students in computing science. It consists of the following chapters: 1. Preliminaries. 2. Categories. 3. Functors. 4. Diagrams, naturality and sketches. 5. Products and sums. 6. Cartesian closed categories. 7. Finite discrete sketches. 8. Limits and colimits. 9. More about sketches. 10. The category of sketches. 11. Fibrations. 12. Adjoints. 13. Algebras for endofunctors. 14. Toposes. There is also a large appendix containing detailed solutions to all the exercises. The authors lay emphasis on examples from and applications to computing science. The book, written in a lively and sometimes provocative style (e.g., Section 14.3 is entitled “Is a two-element poset complete?”), could serve as an introduction to the authors’ monograph “Toposes, triples and theories” (1985; Zbl 0567.18001). The authors managed to write a nice book which is easy to understand and of moderate size, but covers a relatively broad scope of topics.