I am familiar with basic algebraic geometry in the tradition of Hartshorne's book. Discrete valuation rings appear there in the criteria for separatedness/properness, and are used to define the order of a pole/zero of a meromorphic function on a nonsingular variety. Furthermore, points of a nonsingular projective curve are in bijection with valuation rings on of the function field.
In Ravi Vakil's notes on algebraic geometry there is a quote of a letter from Serre to Grothendieck. It appears that Grothendieck disliked valuation rings, whereas Weil thought that valuation rings should play a central role. I have heard from other sources that valued fields have applications in geometry.
My questions are:
1) What is the importance of valuation rings in algebraic geometry other than what I mentioned in the first paragraph? For example, does the topology on a valuation ring play a role? Are valuation rings which are non-discrete important? Can they be used in intersection theory?
2) Is there any old paper, accessible as much as possible to someone who is familiar with the language of schemes, where valuation rings / valued fields are used?