Abstract: In this paper,the concept of simplest filiform Lie algebra is introduced and properties of n-dimensional simplest filiform Lie algebra g are discussed. A filiform Lie algebra g over any field F is referred to as simplest if its Lie multiplication between elements of an adapted basis is trivial except for ones defining the basis. Almost all discussions in the article are under the assumption o dim g≥4. A realization of such a Lie algebra g is given and derivation algebra Der g is determined. It is proved that if for the base field F, its characteristic is zero or p＞n-2 the derivation algebra Der g is a solvable complete Lie algebra.Automorphism groups of simplest filiform Lie algebra are determined and it is proved that it is proved that they are centerless and solvable if ∣F∣≥n. At the end of the paper ,it is shown that the simplest filiform moludar Lie algebra g is restrictable if and only if n＜p+2. And for the nonrestricable case a minimal p-envelope of g is given.

Key words: complete Lie algebra, modular Lie algebra , filiform Lie algebra, complete Lie algebra, modular Lie algebra