Geometric realizations of the Bethe ansatz equations

These lecture notes are devoted to the recent progress in the geometric aspects of quantum integrable systems based on quantum groups solved using the Bethe ansatz technique. One part is devoted to their enumerative geometry realization through the quantum K-theory of Nakajima quiver varieties. The other part describes a recently studied $q$-deformation of the correspondence between oper connections and Gaudin models. The notes are based on a minicourse at C.I.M.E. Summer School ``Enumerative geometry, quantisation and moduli spaces," September 04-08, 2023.