Some background: I am in the process of writing a research paper for an undergraduate abstract algebra course. I've chosen to write my paper on valuation rings and discrete valuation rings. The goal of the paper is to broaden my own and my classmates' understanding of abstract algebra by independently researching a topic not covered in the course. So far, my paper consists of surveying the properties of valuation rings and giving examples of valuation rings.
What I would like to know: I would like to be able to comment on how valuation rings are utilized in various fields of mathematics. Thus far I've had a hard time finding examples that are both explicit and accessible to me, as all of the literature I've found on valuation rings have been graduate texts. It seems like valuation rings are often used in number theory and algebraic geometry, but how are they applied in those fields? What other fields find valuation rings of significant usefulness, and how are they applied? I would also find so-called real world applications useful for my understanding of the topic, but personally I'm more interested in how mathematicians make use of them.
Assume I know linear and abstract algebra at an undergraduate level. Do not assume I know very much about geometry, number theory, or analysis. I'm currently taking a differential geometry course, but what I'm learning seems entirely separate from anything I've seen relating valuation rings to geometry.
My apologies for the broad question. Please let me know if I can clarify or specify in any way.