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$\mathcal{O}$-operators are important in broad areas in mathematics andphysics, such as integrable systems, the classical Yang-Baxter equation,pre-Lie algebras and splitting of operads. In this paper, a deformation theoryof $\mathcal{O}$-operators is established in consistence with the generalprinciples of deformation theories. On the one hand, $\mathcal{O}$-operatorsare shown to be characterized as the Maurer-Cartan elements in a suitablegraded Lie algebra. A given $\mathcal{O}$-operator gives rise to a differentialgraded Lie algebra whose Maurer-Cartan elements characterize deformations ofthe given $\mathcal{O}$-operator. On the other hand, a Lie algebra with arepresentation is identified from an $\mathcal{O}$-operator $T$ such that thecorresponding Chevalley-Eilenberg cohomology controls deformations of $T$, thuscan be regarded as an analogue of the Andr\'e-Quillen cohomology for the$\mathcal{O}$-operator. Thereafter, infinitesimal and formal deformations of$\mathcal{O}$-operators are studied. In particular, the notion of Nijenhuiselements is introduced to characterize trivial infinitesimal deformations.Formal deformations and extendibility of order $n$ deformations of an$\mathcal{O}$-operator are also characterized in terms of the new cohomologytheory. Applications are given to deformations of Rota-Baxter operators ofweight 0 and skew-symmetric $r$-matrices for the classical Yang-Baxterequation. For skew-symmetric $r$-matrices, there is an independentMaurer-Cartan characterization of the deformations as well as an analogue ofthe Andr\'e-Quillen cohomology controlling the deformations, which turn out tobe equivalent to the ones obtained as $\mathcal{O}$-operators associated to thecoadjoint representations. Finally, infinitesimal deformations ofskew-symmetric $r$-matrices and their corresponding triangular Lie bialgebrasare studied.

Deformations and their controlling cohomologies of $\mathcal{O}$-operators