Let N_(c1).c_n be the variety of polynilpotent Lie algebras of class row (c_1,., c_n). The paper is devoted to present the concepts of polynilpotent multiplier M(c_1.c _n)(L) and cover of a Lie algebra L with respect to the variety N _(c1),.c_n, and compute M(c_1.c_n)(L) for some known Lie algebras. Also, we determine the structure of all covers of Lie algebras whose polynilpotent multipliers are finite-dimensional, and investigate some common properties between covers.
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