This lecture is #4 in a series of 10 lectures:  Quantum groups, which originated in mathematical physics in the study of solvable 1+1 dimensional models, are noncocommutative Hopf algebra deformations of the universal enveloping of a Lie algebra.The adjective "quantum" in quantum cohomology has a very different meaning and origin. Quantum cohomology is a commutative deformation of the algebra structure on the cohomology of an algebraic variety X that takes into account the enumerative geometry of rational curves in X. In recent years, a connection between quantum cohomology of certain very special algebraic varieties and quantum groups has been suggested by Nekrasov and Shatashvili, and from a very different angle by Bezrukavnikov and his collaborators. A large portion of these conjectures have since then been proven. My plan in these lectures is to give a slow-paced introduction to this circle of ideas.