The object of the paper under review is to present some vector-valued Banach-Stone theorems. Let

$X$,

$Y$ be compact Hausdorff spaces,

$E$ and

$F$ two Banach spaces and

$C(X,E)$ the Banach space of all continuous

$E$-valued functions defined on

$X$. A linear map

$T:C(X,E)\to C(Y,F)$ is called separating if

$\parallel f\left(x\right)\parallel \phantom{\rule{4pt}{0ex}}\parallel g\left(x\right)\parallel =0$ for every

$x\in E$ implies that

$\parallel \left(Tf\right)\left(y\right)\parallel \phantom{\rule{4pt}{0ex}}\parallel \left(Tg\right)\left(y\right)\parallel =0$ for every

$y\in Y$. The authors show that every biseparating linear bijection

$T$ (that is, a

$T$ for which

$T$ and

${T}^{-1}$ are separating) is a “weighted composition operator”. This means that there exists a function

$h$ of

$Y$ into the set of all bijective linear maps of

$E$ onto

$F$ and a homeomorphism

$\phi $ from

$Y$ onto

$X$ such that

$Tf\left(y\right)=h\left(y\right)\left(f\right(\phi \left(y\right))$ for every

$f\in C(X,E)$ and

$y\in Y$. It is also shown that

$T$ is bounded if and only if for every

$y\in Y$,

$h\left(y\right)$ is a bounded linear operator from

$E$ onto

$F$. An example of an unbounded

$T$ is given.