TITLE: What are examples illustrating the usefulness of Krull (i.e., rank > 1) valuations?
QUESTION [20 upvotes]: 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?

REPLY [2 votes]: For a different flavor from the answers to date: valuations beyond the classical discrete valuations arising from absolute values have proved extremely useful in analysing structures of certain division algebras finite-dimensional over their centers.  Division algebras (even when finite-dimensional over their centres) are notoriously difficult to get a handle on: for instance, it is almost always impossible to describe their subfields.  However, when the centre $F$ is Henselian with respect to a general valuation (Henselian simply means that Hensel's lemma holds: this substitutes for the notion of completeness that is used in the classical case), then it is easy to see that the valuation on $F$ extends uniquely to $D$, and if further the division algebra $D$ is "tame" (i.e., the characteristic of $\overline{F}$ does not divide $[D:F]$), then one can use the valuations on $F$ and $D$ to obtain significant information about $D$.  
As an example: if further $D$ is totally ramified over $F$, i.e., $|V(D)/V(F)| = [D:F]$, where $V()$ stands for the value group, then it turns out that there is a nondegenerate alternating pairing from $V(D)/V(F) \times V(D)/V(F)$ to the group of $m$-th roots of unity in $\overline{F}$, where $m$ is the period of $D$ in the Brauer group of $F$.  This pairing completely determines $D$: $D$ decomposes as a tensor product of "symbol algebras" (i.e., generalized quaternions), where the tensor factors correspond to a symplectic base for $V(D)/V(F)$.  The isomorphism classes of $F$-subalgebras of $D$ are determined by subgroups of $V(D)/V(F)$, and the $F$-isomorphism classes of subfields of $D$ are determined by totally isotropic subgroups of $V(D)/V(F)$.  This in turn completely determines  tame division algebras over fields of the form $k((x_1))\cdots ((x_n))$ where $k$ is separably closed.
For a survey see the paper: Wadsworth, A. R., Valuation theory on finite dimensional division algebras. Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 385--449, Fields Inst. Commun., 32, Amer. Math. Soc., Providence, RI, 2002. (Math Reviews: MR1928379 (2003g:16023) )