The Kadomtsev–Petviashvili (KP) equation is an important nonlinear PDE in the theory of integrable systems, with rich families of solutions arising both from algebraic geometry and from combinatorics. On one hand, Krichever showed how to build solutions from algebraic curves using Riemann theta functions. On the other, Kodama and Williams connected soliton solutions to the geometry of the positive Grassmannian. In this talk I will describe recent and ongoing work with various collaborators, where we study what happens when algebraic curves degenerate tropically. In this limit, theta-function solutions collapse to soliton solutions, and we can track how the geometry of the tropical curve manifests in the combinatorial structure of the soliton. This provides a new bridge between the algebro-geometric and combinatorial approaches to KP solutions.