This paper is dedicated to describe the linear biseparating maps defined between spaces of vector-valued continuous functions on realcompact spaces. There are thereby generalized to the vector-valued setting several results previously proved by the author, with

*E. Beckenstein* and

*L. Narici* [J. Math. Anal. Appl. 192, No. 1, 258-265 (1995;

Zbl 0828.47024)] and with

*J. J. Font* [Proc. Edinb. Math. Soc., II. Ser. 43, No. 1, 139-147 (2000;

Zbl 0945.46032)], for biseparating maps defined between spaces of scalar-valued functions. Assume that

$X$ is a completely regular Hausdorff space and

$A$ is subring of

$C\left(X\right)$ which separates each point of

$X$ from each point of

$\beta X$. In

$\beta X$ the author defines the equivalence relation

$\sim $, defined as

$x\sim y$ whenever

${f}^{\beta X}\left(x\right)={f}^{\beta X}\left(y\right)$ for every

$f\in A$. In this way, the quotient space

$\gamma X:=\beta X/\sim $ is obtained. The space

$\gamma X$ is a compactification of

$X$ (the Samuelcompactification of

$X$ associated by the uniformity generated by

$A$), and every

$f\in A$ can be continuously extended to a map from

$\gamma X$ into

$\mathbb{K}\cup \left\{\infty \right\}$. The symbolism

$C(X,E)$ (resp.

${C}^{*}(X,E)$) denotes the space of continuous (resp. and bounded) functions on

$X$ taking values in

$E$. Suppose that

$A(X,E)\subset C(X,E)$ is an

$A$-module, where

$A$ is a subring of

$X$ which separates each point of

$X$ from each point of

$\beta X$. The author says that

$A(X,E)$ is compatible with

$A$ if, there exists

$f\in A(X,E)$ with

$f\left(x\right)\ne 0$, and if, given any points

$x,y\in \beta X$ with

$x\sim y$, we have

${\parallel f\parallel}^{\beta X}\left(x\right)={|f\parallel}^{\beta X}\left(y\right)$ for every

$f\in A(X,E)$. A subring

$A\subset C\left(X\right)$ is said to be strongly regular if given

${x}_{0}\in \gamma X$ and a nonempty closed subset

$K$ of

$\gamma X$ which does not contain

${x}_{0}$, there exists

$f\in A$ such that

${f}^{\gamma X}\equiv 1$ on a neigbourhood of

${x}_{0}$ and

${f}^{\gamma X}\left(K\right)\equiv 0$. A map

$T:A(X,E)\to A(Y,F)$ is said to be separating if it is additive and

$\parallel \left(Tf\right)\left(y\right)\parallel \xb7\parallel \left(Tg\right)\left(y\right)\parallel =0$ for all

$y\in Y$ whenever

$f,g\in A(X,E)$ satisfy

$\parallel f\left(x\right)\parallel \xb7\parallel g\left(x\right)\parallel =0$ for all

$x\in X$. Moreover,

$T$ is said to be biseparating if it is bijective and both

$T$ and

${T}^{-1}$ are separating. Among others, the following main results are obtained: (1) Suppose that

$A(X,E)\subset C(X,E)$ and

$A(Y,F)\subset C(Y,F)$ are an

$A$-module and a

$B$-module compatible with

$A$ and

$B$, respectively, where

$A\subset C\left(X\right)$ and

$B\subset C\left(Y\right)$ are strongly regular rings. Also, in case when

$\gamma X\ne \beta X$ and

$\gamma Y\ne \beta Y$, it is assumed that for every

$x\in \beta X$ and

$y\in \beta Y$, there exists

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

$g\in A(Y,F)$ satisfying

${\parallel f\parallel}^{\beta X}\left(x\right)\ne 0$ and

${\parallel g\parallel}^{\beta Y}\left(y\right)\ne 0$. If

$T:A(X,E)\to A(Y,F)$ is a biseparating map, then

$\gamma X$ and

$\gamma Y$ are homeomorphic.(2) If

$T:{C}^{*}(X,E)\to {C}^{*}(Y,F)$ is biseparating, then

$\upsilon X$ and

$\upsilon Y$ are homeomorphic.