Reaction diffusion equations have been introduced during the early 20th century to model the density of populations undergoing range expansions in various contexts. These equations commonly admit travelling wave solutions, i.e. the population expands at a constant speed with a stationary profile. These deterministic models can be obtained as rescaling limits of stochastic population models when the population density tends to infinity. But do these stochastic models also admit such (random) travelling fronts? If so, what is the asymptotic speed of these fronts, and how does the nature of the front affect this speed? These questions have been the subject of many studies in the case of the Fisher-Kolmogorov-Petrovsky-Piskunov equation, and in this talk I will give some partial answers in the case of reaction-diffusion equations with a bistable reaction term. The latter type of equations arises when one is interested in the motion of hybrid zones or the expansion of populations with an Allee effect. We shall see that their behaviour is in sharp contrast with that of the stochastic F-KPP equation.