  We develop a theory of universal central extensions of Hom-Lie algebras.
Classical results of universal central extensions of Lie algebras cannot be
completely extended to Hom-Lie algebras setting, because of the composition of
two central extensions is not central. This fact leads to introduce the notion
of universal $\alpha$-central extension. Classical results as the existence of
a universal central extension of a perfect Hom-Lie algebra remains true, but
others as the central extensions of the middle term of a universal central
extension is split only holds for $\alpha$-central extensions. A homological
characterization of universal ($\alpha$)-central extensions is given.
