  These lecture notes concern the semi-holomorphic 4d Chern-Simons theory and
its applications to classical integrable field theories in 2d and in particular
integrable sigma-models. After introducing the main properties of the
Chern-Simons theory in 3d, we will define its 4d analogue and explain how it is
naturally related to the Lax formalism of integrable 2d theories. Moreover, we
will explain how varying the boundary conditions imposed on this 4d theory
allows to recover various occurences of integrable sigma-models through this
construction, in particular illustrating this on two simple examples: the
Principal Chiral Model and its Yang-Baxter deformation. These notes were
written for the lectures delivered at the school "Integrability, Dualities and
Deformations", that ran from 23 to 27 August 2021 in Santiago de Compostela and
virtually.
