The classical result of K. De Leeuw (Ann. Math., 1965) states that for 1 < p < ∞, given a continuous and bounded function which defines a Fourier multiplier on L^p, the sequence defined by its restriction to the integers defines a Fourier multiplier in L^p in the periodic case. In the first part of the talk, we will give an idea of the proof of De Leeuw’s result, using the Transference method approach developed by R. Coifman and G. Weiss, which only uses the underlying group structure. We will point out the main ingredients of the proof, as well as the main obstacles for generalisations. The study of multilinear multipliers is motivated by their appearance in analysis, such as in the work of R. Coifman and Y. Meyer on singular integral operators and commutators. The proof of M. Lacey and C. Thiele on the boundedness of the bilinear Hilbert transform, ignited interest in questions related to multilinear operators, which lead to the study of the validity of multilinear counterparts to classical linear results. In particular, De Leeuw’s type results have been also obtained in the context of multilinear multipliers. In the second part of the talk, we will present an abstract version of a De Leeuw’s type result for multilinear multipliers on LCA groups.