  In this paper, we study compatible Leibniz algebras. We characterize
compatible Leibniz algebras in terms of Maurer-Cartan elements of a suitable
differential graded Lie algebra. We define a cohomology theory of compatible
Leibniz algebras which in particular controls a one-parameter formal
deformation theory of this algebraic structure. Motivated by a classical
application of cohomology, we moreover study the abelian extension of
compatible Leibniz algebras.
