Studies for the classical Yang-Baxter equation and the associative Yang-Baxter equation that can be tracked back to the works of Semonov-Tian-Shansky and Kupershmidt on Rota-Baxter Lie algebras and O-operators. In this talk, we show that solutions of the associative Yang-Baxter equation are characterized in terms of generalized O-operators, and in terms of the classical O-operators precisely when the solutions satisfy an invariant condition. When the invariant condition is compatible with a Frobenius algebra, such solutions have a close relationship with Rota-Baxter operators on the Frobenius algebra. In particular, some developments of the Yang-Baxter equation are also given. This talk is based on joint works with professors Chengming Bai, Xing Gao, Li Guo, and Dr Xiaosong Peng.