In this short chapter, we discuss Schur multipliers restricted to various subspaces \(E \subset B(H)\). We first discuss the case when \(H = \ell_2\) and E is the sub-class of all Hankel matrices. We show that the Schur multipliers which are completely bounded maps from E to E are closely related to the Fourier multipliers on the Hardy space H1. Analogously, when \(H = \ell_2(G)\) and E is the reduced C*-algebra \(C_{\lambda}^{*}(G)\), then the Schur multipliers which are completely bounded maps from E to E are identical to the completely bounded multipliers of \(C_{\lambda}^{*}(G)\) or equivalently to the (so-called) Herz-Schur multipliers of G.