Understanding and predicting the spatial spread of emerging pathogens is a major challenge for the public health management of infectious diseases. Theoretical epidemiology shows that the speed of an epidemic is governed by the life-history characteristics of the pathogen and its ability to disperse. Rapid evolution of these traits during the invasion may thus affect the speed of epidemics. Here we study the influence of virulence evolution on the spatial spread of an epidemic. At the edge of the invasion front, we show that more virulent and transmissible genotypes are expected to win the competition with other pathogens. Behind the front line, however, more prudent exploitation strategies outcompete virulent pathogens. Crucially, even when the presence of the virulent mutant is limited to the edge of the front, the invasion speed can be dramatically altered by pathogen evolution. We support our analysis with individual-based simulations and we discuss the additional effects of demographic stochasticity taking place at the front line on virulence evolution. We confirm that an increase of virulence can occur at the front, but only if the carrying capacity of the invading pathogen is large enough. These results are discussed in the light of recent empirical studies examining virulence evolution at the edge of spreading epidemics.