In this paper, first we introduce the notion of quadratic Rota-Baxter Lie algebras of arbitrary weight, and show that there is a one-to-one correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras of nonzero weight. Then we introduce the notions of matched pairs, bialgebras and Manin triples of Rota-Baxter Lie algebras of arbitrary weight, and show that Rota-Baxter Lie bialgebras, Manin triples of Rota-Baxter Lie algebras and certain matched pairs of Rota-Baxter Lie algebras are equivalent. The coadjoint representations and quadratic Rota-Baxter Lie algebras play important roles in the whole study. Finally we generalize some results to the Lie group context. In particular, we show that there is a one-to-one correspondence between factorizable Poisson Lie groups and quadratic Rota-Baxter Lie groups. Contents 1. Introduction 1 2. Factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras 3 2.1. Preliminaries on factorizable Lie bialgebras 4 2.2. Factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras 5 3. Matched pairs of Rota-Baxter Lie algebras 8 3.1. Rota-Baxter Lie algebras and matched pairs of Lie algebras 8 3.2. Matched pairs of Rota-Baxter Lie algebras 11 4. Rota-Baxter Lie bialgebras 14 4.1. Rota-Baxter Lie bialgebras and matched pairs of Rota-Baxter Lie algebras 15 4.2. Rota-Baxter Lie bialgebras and Manin triples of Rota-Baxter Lie algebras 17 5. Rota-Baxter Lie groups and factorizable Poisson Lie groups 19 5.1. Factorizable Poisson Lie groups and quadratic Rota-Baxter Lie groups 19 5.2. Rota-Baxter Lie groups and matched pairs of Lie groups 22 References 25