•Regular variation is a powerful tool for analyzing fat-tailed invasions.•Tail indices control rates of acceleration of fat-tailed invasions.•Our analyses apply to populations with either no or weak Allee effects.•Initial conditions also affect fat-tailed invasion rates.•Measuring accelerating invasions requires new tools and methods. Display omitted Integrodifference equations (IDEs) are used in ecology to model the growth and spatial spread of populations. With IDEs, dispersal is specified with a probability density function, called the dispersal kernel, and the shape of the kernel influences how rapidly invasions progress. In this paper, we apply tail additivity, a property of regularly varying probability densities, to model invasions with fat-tailed (power-law decay) dispersal in one dimension. We show that fat-tailed invasions progress geometrically fast, with the rate of spread depending on the degree of fatness of the tails. Our analyses apply to populations with no Allee effect as well as weak Allee effects, and we conduct simulations to show that fat-tailed invasions with weak Allee effects produce accelerating invasions. We analyze point-release and front-release invasions, corresponding to newly-established and well-established populations, and we find that front-release invasions gain a permanent speed-up over point-release invasions, invading at a faster geometric rate that persists for all time. Since accelerating invasions are qualitatively different than constant-speed invasions, we also discuss how measures of invasion must be modified and reconsidered when invasions accelerate.