0. Preliminaries.- 1. Definitions.- 2. First properties.- 3. Good and bad actions.- 4. Further properties.- 5. Resumé of some results of Grothendieck.- 1. Fundamental theorems for the actions of reductive groups.- 1. Definitions.- 2. The affine case.- 3. Linearization of an invertible sheaf.- 4. The general case.- 5. Functional properties.- 2. Analysis of stability.- 1. A numeral criterion.- 2. The flag complex.- 3. Applications.- 3. An elementary example.- 1. Pre-stability.- 2. Stability.- 4. Further examples.- 1. Binary quantics.- 2. Hypersurfaces.- 3. Counter-examples.- 4. Sequences of linear subspaces.- 5. The projective adjoint action.- 6. Space curves.- 5. The problem of moduli - 1st construction.- 1. General discussion.- 2. Moduli as an orbit space.- 3. First chern classes.- 4. Utilization of 4.6.- 6. Abelian schemes.- 1. Duals.- 2. Polarizations.- 3. Deformations.- 7. The method of covariants - 2nd construction.- 1. The technique.- 2. Moduli as an orbit space.- 3. The covariant.- 4. Application to curves.- 8. The moment map.- 1. Symplectic geometry.- 2. Symplectic quotients and geometric invariant theory.- 3. Kähler and hyperkähler quotients.- 4. Singular quotients.- 5. Geometry of the moment map.- 6. The cohomology of quotients: the symplectic case.- 7. The cohomology of quotients: the algebraic case.- 8. Vector bundles and the Yang-Mills functional.- 9. Yang-Mills theory over Riemann surfaces.- Appendix to Chapter 1.- Appendix to Chapter 2.- Appendix to Chapter 3.- Appendix to Chapter 4.- Appendix to Chapter 5.- Appendix to Chapter 7.- References.- Index of definitions and notations.

¿Geometric Invariant Theory¿ by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged edition appeared in 1982) is the standard reference on applications of invariant theory to the construction of moduli spaces. This third, revised edition has been long awaited for by the mathematical community. It is now appearing in a completely updated and enlarged version with an additional chapter on the moment map by Prof. Frances Kirwan (Oxford) and a fully updated bibliography of work in this area. The book deals firstly with actions of algebraic groups on algebraic varieties, separating orbits by invariants and construction quotient spaces; and secondly with applications of this theory to the construction of moduli spaces. It is a systematic exposition of the geometric aspects of the classical theory of polynomial invariants.