Traveling waves arise in partial differential equations in a broad range of applications. The notion of stability of a traveling wave solution concerns its resilience to small perturbations and can frequently be inferred from an eigenvalue problem obtained by linearizing the PDE about the solution. I will discuss these ideas in the context of the FitzHugh--Nagumo system, a simplified model of nerve impulse propagation. I will present existence and stability results for (multi)pulse solutions, and I will describe a phenomenon whereby unstable eigenvalues accumulate as a single pulse is continuously deformed into a double pulse.