  This review article discusses recent progress in understanding of various
families of integrable models in terms of algebraic geometry, representation
theory, and physics. In particular, we address the connections between soluble
many-body systems of Calogero-Ruijsenaars type, quantum spin chains, spaces of
opers, representations of double affine Hecke algebras, enumerative counts to
quiver varieties, to name just a few. We formulate several conjectures and open
problems. This is a contribution to the proceedings of the conference on
Elliptic Integrable Systems and Representation Theory, which was held in August
2023 at University of Tokyo.
