The ODE/IM correspondence is a conjectural relation between a class of affine Lie algebra-valued connections — leading to systems of linear ODEs — and certain algebraic equations, known as Bethe Ansatz and related to quantum Integrable Models. In the simplest case of the algebra sl₂, the related ODE is a Schroedinger equation and the corresponding Bethe Ansatz is (strictly related with) the vacuum eingenvalue of the quantum KdV equation. In this talk, I will describe the results achieved in in collaboration with Davide Masoero and Daniele Valeri in the recent paper http://arxiv.org/abs/1501.07421, where we proved the ODE/IM correspondence in the case of simply-laced simple Lie algebras. The techniques used for the proof include the representation theory of simple Lie algebras and affine Kac-Moody algebras, as well as the asymptotic theory of linear ODEs in the complex plane.