The main subject of this book is singular homology, the simplest instrument of algebraic topology. In studying this theory and its applications, the book also investigates parts of other disciplines that form its underlying structural layout: homological algebra, homotopy theory, and category theory. The cohomology theories of Alexander-Spanier, ¿ech, and de Rham are also presented. This book is an introduction to a complex domain, with references to its advanced parts and ramifications. It is written with minimal prerequisites - basic general topology and little else - and a moderate progression, starting with a very elementary beginning. A consistent part of the exposition is organised in the form of exercises, with suitable hints and solutions. The book can be used as a textbook for a first course in algebraic topology or for self-study, and also as a reference book for further study. It is based on a previous version, which has been entirely reviewed, adding several presentations of advanced subjects, and exploring more deeply the interplay of homology and homotopy.