1 Cohomological Intersection Theory and the Nakai-Moishezon Criterion of Ampleness. - 2 The Hodge Index Theorem and the Structure of the Intersection Matrix of a Fiber. - 3 Criteria of Contractability and Rational Singularities. - 4 Properties of Rational Singularities. - 5 Noether's Formula the Picard Scheme the Albanese Variety and Plurigenera. - 6 Existence of Minimal Models. - 7 Morphisms from a Surface to a Curve. Elliptic and Quasielliptic Fibrations. - 8 Canonical Dimension of an Elliptic or Quasielliptic Fibration. - 9 The Classification Theorem According to Canonical Dimension. - 10 Surfaces with Canonical Dimension Zero (char(k) ? 2 3). - 11 Ruled Surfaces. The Noether-Tsen Criterion. - 12 Minimal Models of Ruled Surfaces. - 13 Characterization of Ruled and Rational Surfaces. - 14 Zariski Decomposition and Applications. - 15 Appendix: Further Reading. - References. Language: English