Let \(G\) be a domain in \(\mathbb{R}^n\), where \(G\) is a proper subset of \(\mathbb{R}^n\). For \(x,y \in G\), the Apollonian metric is defined by \(\alpha_G(x,y)=\sup_{a,b\in \partial G}\log\frac{\mid a-x \mid}{\mid a-y \mid}\frac{\mid b-y \mid}{\mid b-x \mid}\). In this paper the author considers domains in \(\mathbb{R}^n\) and mappings with respect to the Apollonian metric. For a quasidisk \(G\) in \(\mathbb{R}^2\) and an Apollonian bilipschitz mapping \(f:G \to \mathbb{R}^2\), the author shows that the image \(f(G)\) is also a quasidisk. Therefore \(f\) is the restriction of \(g\) to \(G\), where \(g\) is a quasiconformal mapping of \(\mathbb{R}^2\cup \{\infty\}\). For a generalization of the theorem on quasidisks to that on quasiballs, A-uniform domains are treated (it is proved that every quasiball is A-uniform in this paper). For a domain \(G\) in \(\mathbb{R}^n\), the necessary and sufficient conditions for \(G\) to be A-uniform are also given, for example, \(G\) is A-uniform if and only if \(G\) is quasi-isotropic and \(\alpha_G\) is quasiconvex. The author shows that if \(G\) is A-uniform in \(\mathbb{R}^n\) and \(f:G \to \mathbb{R}^n\) is an Apollonian bilipschitz mapping then \(f(G)\) is A-uniform if and only if \(f\) is quasiconformal in \(G\). In the case that \(G\) is a ball in \(\mathbb{R}^n\) and \(f:G\to \mathbb{R}^n\) is an Apollonian isometry, the author shows that \(f(G)\) is a ball and that \(f\) is the restriction of \(g\) to \(G\), where \(g\) is a Möbius mapping of \(\mathbb{R}^n\cup \{\infty\}\). This paper is detailed in a terminological description and is likeable.