The Temperley-Lieb algebra is the simplest of the "diagram algebras", appearing as a basic ingredient of the mathematical structure of spin chains and loop models. It is a finite-dimensional associative algebra, depending on a single parameter $q$, whose representation theory is therefore very important for exploring a wide range of integrable models. One interesting (or annoying, depending on your point of view) feature of the Temperley-Lieb algebra is that the $q$ that are physically relevant are precisely those for which there exist representations which cannot be completely reduced, ie written as a direct sum of irreducibles. In this talk, I'll give an overview of basic Temperley-Lieb constructions, leading up to how a single picture, the Bratteli diagram, encodes pretty much all the representation theory that you need to analyse your integrable models (properly!).