Classical integrable field theories broadly fall into two classes: ultralocal and non-ultralocal ones. While the quantum inverse scattering method provides a general way of quantising ultralocal theories, there is currently no systematic way of quantising non-ultralocal theories. I will review how non-ultralocal classical integrable field theories can be thought of as classical Gaudin models associated with affine Kac-Moody algebra. I will then explain why this change of perspective is expected to (i) provide a general framework for quantising non-utralocal integrable field theories, and (ii) underly the ODE/IM correspondence for these theories by analogy with the well-understood case of Gaudin models associated with finite-dimensional semisimple Lie algebras.