I Methods.- 1. Preliminaries.- 2 Derivations and polynomial automorphisms.- 3 Invertibility criteria and inversion formulae.- 4 Injective morphisms.- 5 The tame automorphism group of a polynomial ring.- 6 Stabilization Methods.- 7 Polynomial maps with nilpotent Jacobian.- II Applications.- 8 Applications of polynomial mappings to dynamical systems.- 9 Group actions.- 10 The Jacobian Conjecture.- III Appendices.- A Some commutative algebra.- A.1 Rings.- A.2 Modules.- A.3 Localization.- A.4 Completions.- A.5 Finiteness conditions and integral extensions.- A.6 The universal coefficients method.- B Some basic results from algebraic geometry.- B.1 Algebraic sets.- B.2 Morphisms of irreducible affine algebraic varieties.- C Some results from Gröbner basis theory.- C.1 Definitions and basic properties.- C.2 Applications: several algorithms.- D Flatness.- D.1 Flat modules and algebras.- D.2 Flat morphisms between affine algebraic varieties.- E.2 Direct and inverse images.- F Special examples and counterexamples.- Authors Index.