Effective results and methods for Diophantine equations over finitely generated domains This book is devoted to Diophantine equations where the solutions are taken from an integral domain of characteristic 0 that is finitely generated over Z that is a domain of the shape Z[z1; : : : ; zr] with quotient field of characteristic 0 where the generators z1; : : : ; zr may be algebraic or transcendental over Q. For instance the ring of integers and the rings of S integers of a number field are finitely generated domains where all generators are algebraic. Our aim is to prove effective finiteness results for certain classes of Diophantine equations i.e. results that not only show that the equations from the said classes have only finitely many solutions but whose proofs provide methods to determine the solutions in principle