I will introduce the Gaudin Model. This model is described by a system of commuting Hamiltonians. I will explain how the eigenvalue equations for these Hamiltonians arise as the critical level limit of the Knizhnik-Zamolodchikov equations. In particular, some eigenvectors can be built from H3 correlators. Then I will use the H3-Liouville relation to relate these correlators to Liouville theory correlators. The critical level limit is interpreted in Liouville theory as a geometrical limit. This leads to the construction of Gaudin eigenvalues from the accessory parameters which arise in the uniformization of certain Riemann surfaces.