Existentially complete structures and existentially universal structures -- Model-completions and model-companions -- Infinite forcing in model theory -- Approximating chains for -- Finite forcing in model theory -- Axiomatizations -- Forcing and recursion theory -- Existentially complete models -- Simple models and R.E. ultrapowers -- Regular models -- Regular models and second order models for arithmetic -- Generic models and the analytic hierarchy -- Applications to complete extensions of peano's arithmetic -- Existentially complete division rings -- Nullstellensatz -- Classes of existentially complete division algebras