1 Curves on a Surface. - Invariants of a surface. - Divisors on a surface. - Adjunction and arithmetic genus. - The Riemann-Roch formula. - Algebraic proof of the Hodge index theorem. - Ample and nef divisors. - Exercises. - 2 Coherent Sheaves. - What is a coherent sheaf?. - A rapid review of Chern classes for projective varieties. - Rank 2 bundles and sub-line bundles. - Elementary modifications. - Singularities of coherent sheaves. - Torsion free and reflexive sheaves. - Double covers. - Appendix: some commutative algebra. - Exercises. - 3 Birational Geometry. - Blowing up. - The Castelnuovo criterion and factorization of birational morphisms. - Minimal models. - More general contractions. - Exercises. - 4 Stability. - Definition of Mumford-Takemoto stability. - Examples for curves. - Some examples of stable bundles on ?2. - Gieseker stability. - Unstable and semistable sheaves. - Change of polarization. - The differential geometry of stable vector bundles. - Exercises. - 5 Some Examples of Surfaces. - Rational ruledsurfaces. - General ruled surfaces. - Linear systems of cubics. - An introduction toK3 surfaces. - Exercises. - 6 Vector Bundles over Ruled Surfaces. - Suitable ample divisors. - Ruled surfaces. - A brief introduction to local and global moduli. - A Zariski open subset of the moduli space. - Exercises. - 7 An Introduction to Elliptic Surfaces. - Singular fibers. - Singular fibers of elliptic fibrations. - Invariants and the canonical bundle formula. - Elliptic surfaces with a section and Weierstrass models. - More general elliptic surfaces. - The fundamental group. - Exercises. - 8 Vector Bundles over Elliptic Surfaces. - Stable bundles on singular curves. - Stable bundles of odd fiber degree over elliptic surfaces. - A Zariski open subset of the moduli space. - An overview of Donaldson invariants. - The 2-dimensional invariant. - Moduli spaces via extensions. - Vector bundles with trivial determinant. - Even fiber degree and multiple fibers. - Exercises. - 9 Bogomolov's Inequality and Applications. - Statement ofthe theorem. - The theorems of Bombieri and Reider. - The proof of Bogomolov's theorem. - Symmetric powers of vector bundles on curves. - Restriction theorems. - Appendix: Galois descent theory. - Exercises. - 10 Classification of Algebraic Surfaces and of Stable. - Bundles. - Outline of the classification of surfaces. - Proof of Castelnuovo's theorem. - The Albanese map. - Proofs of the classification theorems for surfaces. - The Castelnuovo-deFranchis theorem. - Classification of threefolds. - Classification of vector bundles. - Exercises. - References. Language: English