  For a discrete group $G$ and a discrete $G$-space $X$, we identify the
Stone-\v{C}ech compactifications $\beta G$ and $\beta X$ with the sets of all
ultrafilters on $G$ and $X$, and apply the natural action of $\beta G$ on
$\beta X$ to characterize large, thick, thin, sparse and scattered subsets of
$X$. We use $G$-invariant partitions and colorings to define $G$-selective and
$G$-Ramsey ultrafilters on $X$. We show that, in contrast to the
set-theoretical case, these two classes of ultrafilters are distinct. We
consider also universally thin ultrafilters on $\omega$, the $T$-points, and
study interrelations between these ultrafilters and some classical ultrafilters
on $\omega$.
