Hypergraphs and various properties that we can prove about them are the basis of many techniques that are used in modern mathematics. I will mention the proof of Density Hales-Jewett Theorem by Tim Austin. Multidimensional Szmeredi theorem is also be another key-word you might want to look up. The Furstenberg-Katznelson theorem can be proven using hypergraph methods. The mathematics subject classification is 05C65.
And more importantly, take a look at What's new and search for "hypergraphs" to see a lot of other results that involve hypergraph methods in their proofs.
One more thing, for real world applications, hypergraph methods appear in various places including declustering problems which are quite important to scale up the performance of parallel databases.