This paper is devoted to the study of spreading speeds and traveling waves for a class ofreaction-diffusion equations with distributed delay. Such an equation describes growth anddiffusion in a population where the individuals enter a quiescent phase exponentially andstay quiescent for some arbitrary time that is given by a probability density function.The existence of the spreading speed and its coincidence with the minimum wave speed ofmonostable traveling waves are established via the finite-delay approximation approach. Wealso prove the existence of bistable traveling waves in the case where the associatedreaction system admits a bistable structure. Moreover, the global stability and uniquenessof the bistable waves are obtained in the case where the density function has zerotail

We discuss how distributed delays arise in biological models and review theliterature on such models. We indicate why it is important to keep thedistributions in a model as general as possible. We then demonstrate, throughthe analysis of a particular example, what kind of information can be gainedwith only minimal information about the exact distribution of delays.In particular we show that a distribution independent stability region maybe obtained in a similar way that delay independent results are obtained forsystems with discrete delays. Further, we show how approximations to theboundary of the stability region of an equilibrium point may be obtained withknowledge of one, two or three moments of the distribution. We compare theapproximations with the true boundary for the case of uniform and gammadistributions and show that the approximations improve as more moments are used.