This is a basic text on algebraic topology designed for use in a one year course at the masters or beginning Ph.D. level. Basic concepts of the subject like the fundamental group, covering projections, simplicial complexes, and simplicial homology are discussed at length. Singular homology is introduced to give a glimpse of an abstract homology theory in the sense of Elienberg and Steenrod.Interesting applications, examples and exercises are included. An introductory chapter and an appendix dealing with basic topology and algebra make the book self-contained.In this reprint edition several errors have been corrected. Exercises in various chapters have been increased. A small section on Tor and Ext functors has been added in the Appendix for completeness.