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.