Part I. 1. Introduction; 2. Topological and metric structures; 3. Harmonic maps and global structures; 4. Cauchy Riemann operators; 5. Zeta function and heat kernel determinants; 6. The Faddeev-Popov procedure; 7. Determinant bundles; 8. Chern classes of determinant bundles; 9. Gaussian meaures and random fields; 10. Functional quantization of the Høegh-Krohn and Liouville model on a compact surface; 11. Small time asymptotics for heat-kernel regularized determinants; Part II. 1. Quantization by functional integrals; 2. The Polyakov measure; 3. Formal Lebesgue measures; 4. Gaussian integration; 5. The Faddeev-Popov procedure for bosonic strings; 6. The Polyakov measure in non-critical dimension; 7. The Polyakov measure in critical dimension d=26; 8. Correlation functions.