Let$X$be a rearrangement-invariant Banach function space on$R$^{$n$}and let$V$^{1}$X$be the Sobolev space of functions whose gradient belongs to$X$. We give necessary and sufficient conditions on$X$under which$V$^{1}$X$is continuously embedded into BMO or into$L$_{∞}. In particular, we show that$L$_{$n, ∞$}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into BMO and, similarly,$L$_{$n$, 1}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into$L$_{∞}. We further show that$V$^{1}$X$is a subset of VMO if and only if every function from$X$has an absolutely continuous norm in$L$_{$n, ∞$}. A compact inclusion of$V$^{1}$X$into$C$^{0}is characterized as well.