The Fourier algebra A(G)and the Fourier-Stieltjes algebra B(G)are function algebras that occur naturally in harmonic analysis of a locally compact group G.Unless G is a compact group,A(G)is a proper subalgebra of B(G),since functions in A(G)vanish at infinity while B(G)contains the constant functions.Consider the following question: Does the Fourier algebra A(G)coincide with the subalgebra of B(G)consisting of functions vanishing at infinity? This last algebra is sometimes called the Rajchman algebra.The talk will cover previously known results concerning this question.It will also include a theorem giving sufficient conditions for the question to have an affirmative answer.As an application of the theorem we are able to give new examples of groups whose Fourier and Rajchman algebras coincide.In particular,we show that there exist uncountably many non-compact groups with this property.Finally,we compare groups with completely reducible regular representation and groups whose Fourier and Rajchman algebras coincide.For unimodular groups with completely reducible regular representation,we show that the Fourier algebra may in general be strictly smaller than the Rajchman algebra.