  In this paper we describe central extensions of some nilpotent Leibniz
algebras. Namely, central extensions of the Leibniz algebra with maximal index
of nilpotency are classified. Moreover, non-split central extensions of
naturally graded filiform non-Lie Leibniz algebras are described up to
isomorphism. It is shown that $k$-dimensional central extensions ($k\geq 5$) of
these algebras are split.
