  This paper examines whether the concept of an almost-algebraic Lie algebra
developed by Auslander and Brezin in \cite{ab} can be introduced for Leibniz
algebras. Two possible analogues are considered: almost-reductive and
almost-algebraic Leibniz algebras. For Lie algebras these two concepts are the
same, but that is not the case for Leibniz algebras, the class of
almost-algebraic Leibniz algebras strictly containing that of the
almost-reductive ones. Various properties of these two classes of algebras are
obtained, together with some relationships to $\phi$-free, elementary,
$E$-algebras and $A$-algebras.
