While the theory of nonlinear waves in partial differential equations is very well developed, understanding travelling waves in systems posed on lattices is challenging, and many basic questions remain open. Indeed, travelling waves on lattices can be found only by solving functional differential equations of mixed mode, which are ill-posed as initial-value problems. In addition, propagation failure or pinning occurs frequently for waves with small speeds, which makes it hard to find such waves using perturbation arguments. In this talk, I will outline work on the existence and stability of travelling waves for the discrete FitzHugh-Nagumo system using geometric singular perturbation theory and, if time permits, for weak shocks in semidiscrete systems of conservation laws.