Direct interactions among species are only possible if there is some overlap in their geographical distributions. However, despite intense focus of macroecological research on species geographical ranges, relatively little theoretical and empirical work has been done on the evolution of range overlap. In this study we explore a simple model of range overlap based on a log-normal distribution of species range sizes along a one-dimensional domain, with or without absorbing boundary conditions. In particular, we focus on the mean and variance of range overlap distributions, as well as the topology of the resulting overlap networks with respect to their degree distribution, evenness, and betweenness scores. According to the model, there is an approximately linear relationship between many aspects of the distribution of range overlaps and their underlying species distributions, such as their mean and variance. However, the expected mean number of non-zero range overlaps for a given species varied from linear to convex depending on the variance of the underlying geographical range distribution. The expected topology of range overlap networks varied substantially depending on the mean and variance in the corresponding geographical distributions, particularly in the case of the degree and closeness distributions. Finally, we test the expectations of our model against five datasets of altitudinal distributions of Neotropical birds. We found strong departures from the expectations based on our model, which could potentially result from phylogenetic niche conservatism related to altitudinal gradients in environmental conditions, or from the asymmetric colonization of mountains by species from lowlands. Potential applications of range overlap networks to a variety of ecological and evolutionary phenomena are discussed.