Let $β$ be the functor from Set to CHaus which maps each discrete set X to its Stone-Cech compactification, the set $β$ X of ultrafilters on X. Every graph G with vertex set V naturally gives rise to a graph $βG$ on the set $βV$ of ultrafilters on V . In what follows, we interrelate the properties of G and $βG$. Perhaps the most striking result is that G can be finitely coloured iff $βG$ has no loops.