integrable systems, both classical and quantum, kept reemerging in mathematics and theoretical physics during the past several decades. in this talk, after briefly reviewing classical and quantum integrable systems, i will focus on two recent geometric incarnations of integrable systems based on quantum groups, solved by the algebraic bethe ansatz method. one is motivated by studying 2- and 3-dimensional supersymmetric gauge theories and mathematically explained through enumerative geometry of quiver varieties. another comes from an instance of geometric langlands correspondence. finally, i will explain the relationship between these two geometrizations and discuss their applications.