Abstract: This book gives an account of the present state of research on lattices of elementary substructures and automorphisms of nonstandard models of arithmetic. Major representation theorems are proved, and the important particular case of countable recursively saturated models is discussed in detail. All necessary technical tools are developed. The list includes: constructions of elementary simple extensions; a partial classification of arithmetic types, in particular Gaifman's theory of definable types; forcing in arithmetic; elements of the Kirby-Paris combinatorial theory of cuts; Lascar's generic automorphisms; and applications of Abramson and Harrington's generalization of Ramsey's theorem. There are also chapters discussing 1-like models with interesting second order properties, and a chapter on order types of nonstandard models.

Keywords: nonstandard models, countable recursively saturated models, Gaifman's theory, Kirby-Paris, Lascar's generic automorphisms, Ramsey's theorem