Stable algebraic topology, 1945-1966. J. Peter May Contents: 1. Setting up the foundations 2. The Eilenberg-Steenrod axioms 3. Stable and unstable homotopy groups 4. Spectral sequences and calculations in homology and homotopy 5. Steenrod operations, K(\pi,n)'s, and characteristic classes 6. The introduction of cobordism 7. The route from cobordism towards K-theory 8. Bott periodicity and K-theory 9. The Adams spectral sequence and Hopf invariant one 10. S-duality and the introduction of spectra 11. Oriented cobordism and complex cobordism 12. K-theory, cohomology, and characteristic classes 13. Generalized homology and cohomology theories 14. Vector fields on spheres and J(X) 15. Further applications and refinements of K-theory 16. Bordism and cobordism theories 17. Further work on cobordism and its relation to K-theory 18. High dimensional geometric topology 19. Iterated loop space theory 20. Algebraic K-theory and homotopical algebra 21. The stable homotopy category References