Let \(G\) be a group presented as a quotient of a free group \(F\) by a normal subgroup \(R\). The abelian group \(M^{(c)}(G)=(R\cap\gamma_{c+1}F)/\gamma_{c+1}(R,F)\) (\(c\geq 1\)) where \(\gamma_1F=F\), \(\gamma_{c+1}F=[\gamma_cF,F]\), \(\gamma_1(R,F)=R\), \(\gamma_{c+1}(R,F)=[\gamma_c(R,F),F]\) is called the \(c\)-nilpotent multiplier of \(G\). (The group \(M^{(1)}(G)\) is the Schur multiplier of \(G\).) In this paper \(c\)-nilpotent multipliers of groups are investigated. A lot of results on the structure of \(c\)-nilpotent multipliers is obtained. For many groups \(c\)-nilpotent multipliers are computed.