  This work is intended as an attempt to extend the notion of bialgebra for Lie
algebras to Leibniz algebras and also, the correspondence between the Leibniz
bialgebras and its dual is investigated. Moreover, the coboundary Leibniz
bialgebras, the classical $r$-matrices and Yang-Baxter equations related to the
Leibniz algebras are defined, and some examples are given. Finally, a method
for construction of a dynamical system on a Leibniz manifold via Leibniz
bialgebra is presented.
