Beyond the Hopf algebra, another bialgebra theory on associative algebras is the infinitesimal bialgebra, first discovered by Joni and Rota from combinatorics in the 1970s. In a similar style, Lie bialgebras were introduced by Drinfeld in the early 1980s, evolving into a comprehensive theory including Manin triples, the classical Yang-Baxter equation, relative Rota-Baxter operators (O-operators) and pre-Lie algebras. After a parallel theory for the infinitesimal bialgebra was obtained in the 2000s by M. Aguiar and C. Bai, this Manin triple approach has been applied to give bialgebras for a large class of algebraic structures, and more recently for algebraic structures equipped with uniary operations such as differential and Rota-Baxter operators. This talk gives an introduction to the background and some of these progresses. The talk includes joint works with Chengming Bai and others.