Hypergraphs are mathematical structures that can be used in a very general context to model the relationships between objects in any set. Matchings are then a way of studying those relationships. A huge variety of pure and applied problems can be phrased in terms of matchings in hypergraphs. We are studying and classifying the extreme cases for questions that involve, for example, the size and number of matchings. This will allow us to make definitive statements about what is or is not possible. We will also undertake some probabilistic calculations to determine what is likely or typical. In this way we will develop some important theory that can be applied in the wealth of areas where hypergraphs are used for modelling.