Spatial models of population distributions have traditionally been formulated in terms of partial differential equations with diffusion terms associated with random motion of the organisms. The invasion of a population into a new domain is often described by travelling wave solutions. (A typical example is the Fisher equation.) When a finite population migrates from one area to another, one expects that the appropriate mathematical representation would be that of a travelling pulse. However, when one seeks travelling pulse solutions to reasonable spatial models, severe problems can arise. I will describe one research project motivated by locust swarming behaviour that lead to the realization that such ideal solutions may not be relevant. Some ideas for more realistic models will be discussed. (joint work with James Watmough and Daniel Grunbaum, and more recently, with Alex Mogilner.)