Basing ourselves on Janelidze and Kelly's general notion of central
extension, we study universal central extensions in the context of semi-abelian
categories. Thus we unify classical, recent and new results in one conceptual
framework. The theory we develop is relative to a chosen Birkhoff subcategory
of the category considered: for instance, we consider groups vs. abelian
groups, Lie algebras vs. vector spaces, precrossed modules vs. crossed modules
and Leibniz algebras vs. Lie algebras.