## ABSTRACT

In the chapter we study complex non Lie filiform Leibniz algebras. For Lie algebras, the notion of p-filiformity ( p ∈ N ∪ { 0 } https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780429344336/b8a1f226-c653-40d6-ac67-d9bc131130ff/content/inline-math4_1.jpg"/> ) makes sense for p ≥ 1 and loses sense for p = 0, since a Lie algebra has at least two generators. In the case of Leibniz algebras, this notion is meaningful for p = 0; so the introduction of null-filiform algebra is quite justified. We give some equivalent conditions for a Leibniz algebra to be filiform and describe naturally graded complex filiform Leibniz algebras.