Let

$G$ be a locally compact group. The Fourier-Stieltjes algebra

$B\left(G\right)$ and the Fourier algebra

$A\left(G\right)$ were for the first time investigated by P. Eymard in 1964. The paper proves mainly that for two locally compact amenable groups

${G}_{1}$ and

${G}_{2}$, the Fourier algebras

$A\left({G}_{1}\right)$ and

$A\left({G}_{2}\right)$ are algebra isomorphic if and only if there exists a disjointness preserving bijection between them (Theorem 4), and such disjointness preserving bijection of

$A\left({G}_{1}\right)$ onto

$A\left({G}_{2}\right)$ can be extended, in a unique way, to a weighted composition bijection of

$B\left({G}_{1}\right)$ onto

$B\left({G}_{2}\right)$ (Theorem 5). The author notices that if the amenability of the group

$G$ is dropped, these results may fail to be true. The following may be an interesting question (like the inverse problem of the paper): can an isometric isomorphism or a bipositive algebra isomorphism between

$A\left({G}_{1}\right)$ and

$A\left({G}_{2}\right)$ (or

$B\left({G}_{1}\right)$ and

$B\left({G}_{2}\right))$ be deduced to a topological isomorphism of

${G}_{1}$ onto

${G}_{2}$?