Fourier and Schur multipliers of groups are indispensible in the study of approximation properties and various problems involving non-commutative harmonic analysis. In this talk we introduce L^p-Fourier multipliers for arbitrary groups and study the close relation between such a multiplier and its corresponding Schur multiplier. In particular, we show how to generalize a result by Neuwirth and Ricard stating that for a discrete amenable group, the completely bounded norm of a L^p-Fourier multiplier equals the completely bounded norm of its associated Schur multiplier. We will relate our results to (and stretch their importance for) approximation properties of groups and non-commutative L^p-spaces.