  Schur multipliers were introduced by Schur in the early 20th century and have
since then found a considerable number of applications in Analysis and enjoyed
an intensive development. Apart from the beauty of the subject in itself,
sources of interest in them were connections with Perturbation Theory, Harmonic
Analysis, the Theory of Operator Integrals and others. Advances in the
quantisation of Schur multipliers were recently made by Kissin and Shulman. The
aim of the present article is to summarise a part of the ideas and results in
the theory of Schur and operator multipliers. We start with the classical Schur
multipliers defined by Schur and their characterisation by Grothendieck, and
make our way through measurable multipliers studied by Peller and Spronk,
operator multipliers defined by Kissin and Shulman and, finally,
multidimensional Schur and operator multipliers developed by Juschenko and the
authors. We point out connections of the area with Harmonic Analysis and the
Theory of Operator Integrals.
