For each prime number p, there is a number system — the p-adic number system — rather analogous to the standard (real) number system. This system was discovered in the late 1800s and has played a vital role in many areas of mathematics. The past 15 years have witnessed some major breakthroughs in our understanding of geometries built from this number system; this has had downstream effects, such as the solution of longstanding problems across a variety of fields. In this talk, I will first explain what the p-adic number system is, and then give a high-level overview of some of the new ideas that led to the recent developments.