The main goal of this talk is to present the generalization of   classical results that characterize universal central extensions of Leibniz (Lie) algebras to the framework of Hom-Leinbiz (Hom-Lie) algebras. In the category of Hom-Leibniz (Hom-Lie) algebras we introduce the notion of Hom-co-representation (Hom-L-module) as  adequate coefficients to construct the chain complex  from which we compute the Leibniz (Lie) homology of Hom-Leibniz (Hom-Lie) algebras. Nevertheless, in this generalization fails the key result that claims the composition of central extensions is central as well. This singularity motivates the introduction of new concepts as \alpha-perfect Hom-Leibniz (Hom-Lie) algebra and \alpha-central extension. Then the corresponding characterizations are given. We also provide the recognition criteria for these kinds of universal central extensions. We prove that an \alpha-perfect Hom-Lie algebra  admits a universal \alpha-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both of them. We introduce Hom-actions, semidirect product and establish the equivalence between split extensions and the semi-direct product extension of Hom-Leibniz algebras. We analyze the functorial properties of the universal (\alpha)-central extensions of (\alpha)-perfect Hom-Leibniz algebras. We establish under what conditions an automorphism or a derivation can be lifted in an \alpha-cover and we analyze  the universal \alpha-central extension of the semi-direct product of two \alpha-perfect Hom-Leibniz algebras.