Understanding the stability of solutions to PDEs is important, because it is typically only stable solutions which are observable. For many PDEs in one spatial dimension, stability is well-understood, largely due to a formulation of the problem in terms of so-called spatial dynamics, where one views the single spatial variable as a time-like evolution variable. This allows for many powerful techniques from the theory of dynamical systems to be applied. In higher spatial dimensions, this perspective is not clearly applicable. In this talk, I will discuss recent work that suggests both that the Maslov index could be a important tool for understanding stability when the system has a symplectic structure, particularly in the multi-dimensional setting, and also suggests a possible analogue of spatial dynamics in the multi-dimensional setting.