Part I. Introduction to Operator Spaces: 1. Completely bounded maps; 2. Minimal tensor product; 3. Minimal and maximal operator space structures on a Banach space; 4. Projective tensor product; 5. The Haagerup tensor product; 6. Characterizations of operator algebras; 7. The operator Hilbert space; 8. Group C*-algebras; 9. Examples and comments; 10. Comparisons; Part II. Operator Spaces and C*-tensor products: 11. C*-norms on tensor products; 12. Nuclearity and approximation properties; 13. C*; 14. Kirchberg's theorem on decomposable maps; 15. The weak expectation property; 16. The local lifting property; 17. Exactness; 18. Local reflexivity; 19. Grothendieck's theorem for operator spaces; 20. Estimating the norms of sums of unitaries; 21. Local theory of operator spaces; 22. B(H) * B(H); 23. Completely isomorphic C*-algebras; 24. Injective and projective operator spaces; Part III. Operator Spaces and Non Self-Adjoint Operator Algebras: 25. Maximal tensor products and free products of non self-adjoint operator algebras; 26. The Blechter-Paulsen factorization; 27. Similarity problems; 28. The Sz-nagy-halmos similarity problem; Solutions to the exercises; References.