  We introduce multidimensional Schur multipliers and characterise them
generalising well known results by Grothendieck and Peller. We define a
multidimensional version of the two dimensional operator multipliers studied
recently by Kissin and Shulman. The multidimensional operator multipliers are
defined as elements of the minimal tensor product of several C*-algebras
satisfying certain boundedness conditions. In the case of commutative
C*-algebras, the multidimensional operator multipliers reduce to continuous
multidimensional Schur multipliers. We show that the multipliers with respect
to some given representations of the corresponding C*-algebras do not change if
the representations are replaced by approximately equivalent ones. We establish
a non-commutative and multidimensional version of the characterisations by
Grothendieck and Peller which shows that universal operator multipliers can be
obtained as certain weak limits of elements of the algebraic tensor product of
the corresponding C*-algebras.
