These lecture notes were held in SISSA by prof. Loriano Bonora before his retirement. Table of contents: Chapter 1. Introduction Part I. d=2 Conformal field theories Chapter 2. The BPZ construction Bibliography Chapter 3. Bosonization in 2D Bibliography Part II. Conformal field theory in d dimensions Chapter 4. Conformal algebra representations Chapter 5. Ward identities and null cone method Chapter 6. OPE and bootstrap Bibliography Part III. 2d conformal field theory and string theory Chapter 7. The Polyakov action Chapter 8. Old covariant quantization Chapter 9. The BRST quantization Bibliography Part IV. CFT and related subjects Chapter 10. Conserved, traceless correlators and anomalies