  These lecture notes are based on a blackboard course given at the XVII Modave
Summer School in Mathematical Physics held from 13 -- 17 September 2021 in
Brussels (Belgium), and aimed at Ph.D. students in High Energy Theoretical
Physics. We start with introducing classical integrability in
finite-dimensional systems to set the stage for our main purpose: introducing
two-dimensional classical field theories which are integrable. We focus on
their zero-curvature formulation through the so-called Lax connection, which
ensures the existence of an infinite tower of conserved charges. We then move
on to their Poisson bracket structure, known as the Sklyanin or Maillet
structure, which ensures complete classical integrability. All the concepts
that we encounter will be illustrated with the integrable Principal Chiral
Model, which is the canonical sigma-model that appears (or its generalisations)
on the worldsheet of many string backgrounds. Along the way we briefly comment
on the properties of integrability at the quantum level.
