For many times, I find people working on schemes over DVRs, and prove theorems on such setting. For example, my latest experience is the "semi-stable reduction theorem" by Kempf, Knudsen, Mumford and Saint-Donat: they proved semi-stable reduction theorem first over $\mathbb{C}$, then turned to the situation over DVR.

It is "natural" for me to work over $\mathbb{C}$, or over character $p$, or even over arbitrary schemes. But why people like to choose DVRs in particular? This question would break to two parts:

(1) Is there any technical advantage to work over DVR?

(2) What is the importance to consider this special case? Does this come from the interests of number theory? I have no clue of that.