Recent work of Nisse-Sottile, Hrushovski-Loeser, Ducros, and Giansiracusa-Giansiracusa has demonstrated that valuation rings of rank >1 play an important role in the geometry of analytic and tropical varieties over non-Archimedean valued fields of rank 1. In this talk, I will present recent work with Dhruv Ranganathan in which we prove several foundational results on the geometry of analytic and tropical varieties over higher rank valued fields, and recent work with Max Hully in which we use rank 2 valuations to give a new, non-analytic proof of Rabinoff's theorem on the correspondence between tropical and algebraic intersection multiplicities.