Travelling waves (in particular fronts, pulses and wave trains) are interesting solutions to partial differential equations on one-dimensional unbounded domains. These waves can be computed numerically in a robust fashion as solutions to appropriate boundary-value problems. The issue addressed in this talk is the computation of the spectrum of the PDE, linearized about a travelling wave. One way of computing the spectrum is to truncate the domain to a large interval, to impose boundary conditions at the endpoints, and to then compute the spectrum of the resulting PDE operator, for instance, by discretization. Of interest is then the effect of the boundary conditions on the spectrum. We discuss this issue and show that boundary conditions can change the spectrum quite dramatically. We also outline an alternative approach to the computation of spectra that relies on Evans-function techniques.