In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises. The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers. An introductory textbook suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises. Part I. Some Underlying Geometric Notions: 1. Homotopy and homotopy type 2. Deformation retractions 3. Homotopy of maps 4. Homotopy equivalent spaces 5. Contractible spaces 6. Cell complexes definitions and examples 7. Subcomplexes 8. Some basic constructions 9. Two criteria for homotopy equivalence 10. The homotopy extension property Part II. Fundamental Group and Covering Spaces: 11. The fundamental group, paths and homotopy 12. The fundamental group of the circle 13. Induced homomorphisms 14. Van Kampen's theorem of free products of groups 15. The van Kampen theorem 16. Applications to cell complexes 17. Covering spaces lifting properties 18. The classification of covering spaces 19. Deck transformations and group actions 20. Additional topics: graphs and free groups 21. K(G,1) spaces 22. Graphs of groups Part III. Homology: 23. Simplicial and singular homology delta-complexes 24. Simplicial homology 25. Singular homology 26. Homotopy invariance 27. Exact sequences and excision 28. The equivalence of simplicial and singular homology 29. Computations and applications degree 30. Cellular homology 31. Euler characteristic 32. Split exact sequences 33. Mayer-Vietoris sequences 34. Homology with coefficients 35. The formal viewpoint axioms for homology 36. Categories and functors 37. Additional topics homology and fundamental group 38. Classical applications 39. Simplicial approximation and the Lefschetz fixed point theorem Part IV. Cohomology: 40. Cohomology groups: the universal coefficient theorem 41. Cohomology of spaces 42. Cup product the cohomology ring 43. External cup product 44. Poincar&#233 duality orientations 45. Cup product 46. Cup product and duality 47. Other forms of duality 48. Additional topics the universal coefficient theorem for homology 49. The Kunneth formula 50. H-spaces and Hopf algebras 51. The cohomology of SO(n) 52. Bockstein homomorphisms 53. Limits 54. More about ext 55. Transfer homomorphisms 56. Local coefficients Part V. Homotopy Theory: 57. Homotopy groups 58. The long exact sequence 59. Whitehead's theorem 60. The Hurewicz theorem 61. Eilenberg-MacLane spaces 62. Homotopy properties of CW complexes cellular approximation 63. Cellular models 64. Excision for homotopy groups 65. Stable homotopy groups 66. Fibrations the homotopy lifting property 67. Fiber bundles 68. Path fibrations and loopspaces 69. Postnikov towers 70. Obstruction theory 71. Additional topics: basepoints and homotopy 72. The Hopf invariant 73. Minimal cell structures 74. Cohomology of fiber bundles 75. Cohomology theories and omega-spectra 76. Spectra and homology theories 77. Eckmann-Hilton duality 78. Stable splittings of spaces 79. The loopspace of a suspension 80. Symmetric products and the Dold-Thom theorem 81. Steenrod squares and powers Appendix: topology of cell complexes The compact-open topology.