  In this study, we extend the result on classification of a subclass of
filiform Leibniz algebras in low dimensions to dimensions seven and eight based
on the technique used by Rakhimov and Bekbaev for classification of subclasses
which arise from naturally graded non-Lie filiform Leibniz algebras. The
subclass considered here arises from naturally graded filiform Lie algebras.
This subclass contains the class of filiform Lie algebras and consequently, by
classifying this subclass, we again re-examine the classification of filiform
Lie algebras. Our resulting list of filifom Lie algebras is compared with that
given by Ancoch\'ea-Bermudez and Goze in 1988 and by G\'{o}mez, Jimenez-Merchan
and Khakimdjanov in 1998.
