This course will give a pedagogical introduction to (quantum) integrability, a topic in mathematical physics with applications ranging from experiments in condensed-matter physics to high-energy theory. The aim is to show some highlights of the field, with a glimpse of the underlying algebraic structures, while keeping technicalities to a minimum. The provisional plan of the course is as follows; details can be adjusted to suit the audience. Part I will cover the (standard) basics of integrability, more or less following my lecture notes arXiv:1501.06805: – the Heisenberg spin chain, – the six-vertex model, – the exact characterisation of their spectrum by Bethe ansatz, – an application to alternating-sign matrices (ASM). – Part II is about the (less standard) basics of long-range integrability: – the Haldane–Shastry spin chain, – the quantum Calogero–Sutherland system, – the exact and explicit characterisation of their spectrum.

IPhT