In this seminar, I will review the construction of certain infinite-dimensional Hamiltonian systems called affine Gaudin models and their applications to the study of integrable two-dimensional field theories (i.e. theories possessing an infinite number of conserved quantities / symmetries). These models are built from Kac-Moody currents, which are fields arising as realisations of infinite-dimensional Lie algebras named affine algebras. After reviewing these notions, I will explain how one can build an infinite family of Poisson-commuting conserved charges from these currents, thus showing the (classical) integrability of the model. In a second part, I will describe the relation between affine Gaudin models and a class of two-dimensional field theories called sigma-models, which appear in various domains of physics (such as string theory, condensed matter, conformal models, etc). In particular, I will explain how this formalism can be used to recover previously known examples of integrable sigma-models but also to produce many new ones in a systematic way.