  This paper studies formal deformations and homotopy theory of Rota-Baxter
algebras of any weight. We define an $L_\infty$-algebra, which controls
simultaneous deformations of associative products and Rota-Baxter operators. As
a consequence, we develop a cohomology theory of Rota-Baxter algebras of any
weight and justify it by interpreting lower degree cohomology groups as formal
deformations and abelian extensions. The notion of homotopy Rota-Baxter
algebras is introduced and it is shown that the operad governing homotopy
Rota-Baxter algebras is a minimal model of the operad of Rota-Baxter algebras.
