  This rather heavy definition gives little clue how useful saturated structures are. If every structure had a saturated elementary extension, many of the results of model theory would be much easier to prove. Unfortunately the existence of saturated elementary extensions depends on features of the surrounding universe of sets. There are technical ways around this obstacle, for example using weakenings of the notion of saturation. We have two main ways of constructing elementary extensions with some degree of saturation. One is by ultrapowers, using cleverly constructed ultrafilters. The other is by taking unions of elementary chains, generalising the proof we gave for the upward Löwenheim-Skolem theorem.