Convergence polygons for connections on nonarchimedean curves

This is a survey article on ordinary differential equations over nonarchimedean fields based on the author's lecture at the 2015 Simons Symposium on nonarchimedean and tropical geometry. Topics include: the convergence polygon associated to a differential equation (or a connection on a curve); links to the formal classification of differential equations (Turrittin-Levelt); index formulas for de Rham cohomology of connections; ramification of finite morphisms; relations with the Oort lifting problem on automorphisms of curves. The appendices include some new technical results and an extensive thematic bibliography.