This is a book on singular homology and cohomology with emphasis on products and manifolds. It covers basic homotopy theory and applications of (co-)homology to homotopy, but does not treat general homology or spectral sequences. Čech cohomology is discussed for locally compact subsets of manifolds, with a systematic treatment in an appendix. The book is based on a one-year course in algebraic topology and can serve as a textbook for such a course. For a shorter course, certain chapters and sections can be used. The text assumes knowledge of general topology, abelian group theory, and category theory, though chapter I provides some help with these. Integral homology is treated up to chapter VI, but general coefficients can be introduced with minor adaptations. The book has eight chapters and an appendix, each divided into sections. Definitions, propositions, and formulas are numbered within each section. Exercises are provided to reinforce concepts and highlight further results. Some exercises are marked with a star for difficulty. The author acknowledges the help of colleagues and students in refining the manuscript. The book includes a detailed table of contents, covering topics such as categories, homology of complexes, singular homology, applications to Euclidean space, cellular decomposition, functors of complexes, products, manifolds, and extensions of functors. It also includes a bibliography and subject index.This is a book on singular homology and cohomology with emphasis on products and manifolds. It covers basic homotopy theory and applications of (co-)homology to homotopy, but does not treat general homology or spectral sequences. Čech cohomology is discussed for locally compact subsets of manifolds, with a systematic treatment in an appendix. The book is based on a one-year course in algebraic topology and can serve as a textbook for such a course. For a shorter course, certain chapters and sections can be used. The text assumes knowledge of general topology, abelian group theory, and category theory, though chapter I provides some help with these. Integral homology is treated up to chapter VI, but general coefficients can be introduced with minor adaptations. The book has eight chapters and an appendix, each divided into sections. Definitions, propositions, and formulas are numbered within each section. Exercises are provided to reinforce concepts and highlight further results. Some exercises are marked with a star for difficulty. The author acknowledges the help of colleagues and students in refining the manuscript. The book includes a detailed table of contents, covering topics such as categories, homology of complexes, singular homology, applications to Euclidean space, cellular decomposition, functors of complexes, products, manifolds, and extensions of functors. It also includes a bibliography and subject index.