Certain systems of polynomial equations, known as QQ-systems, appear in surprising corners of geometry and mathematical physics — from the enumerative geometry of quiver varieties to aspects of the (deformed) geometric Langlands program. They also arise in the study of quantum integrable models of spin chain type, linked to quantum groups and Yangians. Specifically, the solutions to the QQ-system equations characterize the spectrum of these integrable models via the so-called Bethe ansatz equations. In this talk, I will give an introduction to quantum groups and integrable models, illustrated by the familiar example of Heisenberg spin chains. I will then explain how methods from tropical geometry − a combinatorial shadow of algebraic geometry − can be effectively used to construct and analyze solutions to QQ-systems. This is a joint work with Anton Zeitlin