Leibniz algebras were introduced by J.-L.Loday. A skew-symmetric Leibniz algebra is a Lie algebra. In this case the Leibniz identity is just the Jacobi identity. This book is devoted to the classification problem of Leibiz algebra in low dimensional cases. There are two sources to get such a classification. The first of them is naturally graded non Lie filiform Leibniz algebras and the other is the naturally graded filiform Lie algebras. Here we consider Leibniz algebras appearing from the naturally graded non Lie filiform Leibniz algebras. It is known that this class of algebras can be split into two subclasses. However, isomorphisms within each class have not been investigated yet. Recently U.D.Bekbaev and I.S.Rakhimov suggested an approach to the isomorphism problem of Leibniz algebras based on algebraic invariants. This book presents an implementation of this invariant approach in 9- dimensional case. We give the list of all 9- dimensional non Lie filiform Leibniz algebras arising from the naturally graded non Lie filiform Leibniz algebras. The isomorphism criteria and the list of algebraic invariants will be given.