  We describe all central extensions of all $3$-dimensional non-zero complex
Zinbiel algebras. As a corollary, we have a full classification of
$4$-dimensional non-trivial complex Zinbiel algebras and a full classification
of $5$-dimensional non-trivial complex Zinbiel algebras with $2$-dimensional
annihilator, which gives the principal step in the algebraic classification of
$5$-dimensional Zinbiel algebras.
