\"The text which follows is based mostly on lectures at PrincetonUniversity in 1957. The senior author wishes to apologize for the delayin publication.The theory of characteristic classes began in the year 1935 with almostsimultaneous work by HASSLER WHITNEY in the United States andEDUARD STIEFEL in Switzerland. StiefeI\'s thesis, written under thedirection of Heinz Hopf, introduced and studied certain \"\"characteristic\"\"homology classes determined by the tangent bundle of a smooth manifold.Whitney, then at Harvard University, treated the case of an arbitrary spherebundle. Somewhat later he invented the language of cohomology theory,hence the concept of a characteristic cohomology class, and proved thebasic product theorem.\"

\"Preface §1. Smooth Manifolds §2. Vector Bundles §3. Constructing New Vector Bundles Out of Old §4. Stiefel-Whitney Classes §5. Grassmann Manifolds and Universal Bundles §6. A Cell Structure for Grassmann Manifolds §7. The Cohomology Ring H*(Gn; Z/2) §8. Existence of Stiefel-Whitney Classes §9. Oriented Bundles and the Euler Class §10. The Thorn Isomorphism Theorem §11. Computations in a Smooth Manifold §12. Obstructions §13. Complex Vector Bundles and Complex Manifolds §14. Chern Classes §15. Pontrjagin Classes §16. Chern Numbers and Pontrjagin Numbers §17. The Oriented Cobordism Ring Ω* §18. Thorn Spaces and Transversality §19. Multiplicative Sequences and the Signature Theorem §20. Combinatorial Pontrjagin Classes Epilogue Appendix A: Singular Homology and Cohomology Appendix B: Bernoulli Numbers Appendix C: Connections, Curvature, and Characteristic Classes. Bibliography Index\"