Topological Methods in Algebraic Geometry : Reprint of the 1978 Edition One. Preparatory material § 1. Multiplicative sequences §2. Sheaves §3. Fibre bundles § 4. Characteristic classes Two. The cobordism ring § 5. Pontrjagin numbers § 6. The ring $$\\tilde \\Omega \\otimes \\mathcal{Q}$$ Q § 7. The cobordism ring  § 8. The index of a 4 k dimensional manifold § 9. The virtual index Three. The Todd genus § 10. Definition of the Todd genus § 11. The virtual generalised Todd genus § 12. The T characteristic of a GL(q C) bundle § 13. Split manifolds and splitting methods § 14. Multiplicative properties of the Todd genus Four. The Riemann Roch theorem for algebraic manifolds § 15. Cohomology of compact complex manifolds § 16. Further properties of the y characteristic § 17. The virtual  y characteristic § 18. Some fundamental theorems of Kodaira § 19. The virtual  y characteristic for algebraic manifolds § 20. The Riemann Roch theorem for algebraic manifolds and complex analytic line bundles §21. The Riemann Roch theorem for algebraic manifolds and complex analytic vector bundles § 26. Integrality theorems for differentiate manifolds A spectral sequence for complex analytic bundles. In recent years new topological methods especially the theory of sheaves founded by J. LERAY have been applied successfully to algebraic geometry and to the theory of functions of several complex variables. H. CARTAN and J. P. SERRE have shown how fundamental theorems on holomorphically complete manifolds (STEIN manifolds) can be for­ mulated in terms of sheaf theory. These theorems imply many facts of function theory because the domains of holomorphy are holomorphically complete. They can also be applied to algebraic geometry because the complement of a hyperplane section of an algebraic manifold is holo­ morphically complete. J. P. SERRE has obtained important results on algebraic manifolds by these and other methods. Recently many of his results have been proved for algebraic varieties defined over a field of arbitrary characteristic. K. KODAIRA and D. C. SPENCER have also applied sheaf theory to algebraic geometry with great success. Their methods differ from those of SERRE in that they use techniques from differential geometry (harmonic integrals etc. ) but do not make any use of the theory of STEIN manifolds. M. F. ATIYAH and W. V. D. HODGE have dealt successfully with problems on integrals of the second kind on algebraic manifolds with the help of sheaf theory. I was able to work together with K. KODAIRA and D. C. SPENCER during a stay at the Institute for Advanced Study at Princeton from 1952 to 1954.