Universality is the somewhat surprising observation that the behavior of quite different models, sometimes even stemming from different branches of mathematics or physics, may be described by the same class of special functions. An example for this phenomenon that has attracted a good deal of research over the past 25 years are the distribution functions arising in random matrix theory. These distributions also appear in a number of seemingly unrelated models in statistical mechanics and can be described by special functions that originate in the theory of integrable systems. We review some of these results and provide a partial explanation of this phenomenon.