In modern valuation theory, one studies not just absolute values on a field, but also **Krull valuations**. The motivation is easy enough:

If $k$ is a field, a **valuation ring** of $k$ is a subring $R$ such that for every $x \in k^{\times}$, at least one of $x, x^{-1}$ is an element of $R$. (It follows of course that $k$ is the fraction field of $R$.) If $| \ |$ is a non-Archimedean norm on a field, then the set $\{x \in k \ | \ |x| \leq 1 \}$ is a valuation ring. However, the converse does not hold, since if $R$ is a valuation ring, then $k^{\times}/R^{\times}$ need not inject into $\mathbb{R}$: rather it is (under a straightforward extension of the divisibility relation on $R$) a totally ordered abelian group. Moreover, a certain formal power series construction shows that for any totally ordered abelian group $\Gamma$, there exists $k$ and $R$ with $k^{\times}/R^{\times} \cong \Gamma$.

My question is this: what are some instances where having the generality of Krull valuations is useful for solving some problem (which is not *a priori* concerned with valuation theory)? How do Krull valuations arise in algebraic geometry?

I can almost remember one example of this. I believe it is possible to give a quick proof of the Lang-Nishimura Theorem -- that having a smooth $k$-rational point is a birational invariant among complete [hmm, valuative criterion!] $k$-varieties. I think I saw this in some of Bjorn Poonen's lecture notes, but I forget where. [Last year at this time, I would have emailed Bjorn. I am trying out this new approach on the theory that Bjorn can reply if he wishes, and if not someone else will surely be eager to tell me the answer.]

Are there other nice examples? Maybe something to do with resolution of singularities?