Many quantum statistical mechanics systems can be represented by loop soups. In some cases it has been proved, in others conjectured, that phase transitions are characterised by changes in the structures of the corresponding loops. There are many such loops and the joint distribution of their lengths is conjectured to always (hence "universality" in the title) converge to the Poisson-Dirichlet distribution in three and higher dimensions.

This talk will start with a definition of loop soups, list some heuristics to determine if a system can be written in terms of loops, and give a sketch of Schramm's proof for universality of a mean field model.