Includes bibliographical references.

Hilbert's tenth problem: what was done and what is to be done ; Undecidability of existential theories of rings and fields: a survey ; Hilbert's tenth problem over number fields, a survey ; Defining constant polynomials ; Decidability and localglobal principles ; Applications of localglobal principles to arithmetic and geometry ; Regularly losed fields ; Skolem density problems over large Galois extensions of global fields ; An effort to prove that the existential theory of bf Q is undecidable ; Topology of Diophantine sets: remarks on Mazur's conjectures ; Diagonal quadratic forms and Hilbert's tenth problem ; Algebraic geometry over four rings and the frontier to tractability ; Some model theory of compact complex spaces ; Double coset decompositions for algebraic groups over [t] ; Zero estimates for polynomials in 3 and 4 variables using orbits and stabilisers Yuri Matiyasevich ; Thanases Pheidas and Karim Zahidi ; Alexandra Shlapentokh ; Mihai Prunescu ; L Darniere ; Laurent MoretBailly ; Joachim Schmid ; Moshe Jarden and Aharon Razon ; Thanases Pheidas ; Gunther Cornelissen and Karim Zahidi ; Paul Vojta ; J Maurice Rojas ; Anand Pillay ; K H Kim and F W Roush ; Curtis D Bennett Lisa K Elderbrock and Andrew M W Glass