  We classify cuts in (totally) ordered abelian groups $\g$ and compute the
coinitiality and cofinality of all cuts in case $\g$ is divisible, in terms of
data intrinsically associated to the invariance group of the cut. We relate
cuts with small extensions of $\g$ in a natural way, which leads to an explicit
construction of a totally ordered real vector space containing realizations of
all cuts. This construction is applied to the problem of classifying all
extensions of the valuation from a given valued field $K$ to the rational
function field $K(x)$.
