- The mysterious ‘fairy circles’ are vegetation-free discs that cover vast areas along the pro-Namib Desert. Despite 30 yr of research their origin remains unknown. Here we adopt a novel approach that focuses on analysis of the spatial patterns of fairy circles obtained from representative 25-ha aerial images of north-west Namibia. We use spatial point pattern analysis to quantify different features of their spatial structures and then critically inspect existing hypotheses with respect to their ability to generate the observed circle patterns. Our working hypothesis is that fairy circles are a self-organized vegetation pattern. Finally, we test if an existing partial-differential-equation model, that was designed to describe vegetation pattern formation, is able to reproduce the characteristic features of the observed fairy circle patterns. The model is based on key-processes in arid areas such as plant competition for water and local resource-biomass feedbacks. The fairy circles showed at all three study areas the same regular spatial distribution patterns, characterized by Voronoi cells with mostly six corners, negative correlations in their size up to a distance of 13 m, and remarkable homogeneity over large spatial scales. These results cast doubts on abiotic gas-leakage along geological lines or social insects as causal agents of their origin. However, our mathematical model was able to generate spatial patterns that agreed quantitatively in all of these features with the observed patterns. This supports the hypothesis that fairy circles are self-organized vegetation patterns that emerge from positive biomass-water feedbacks involving water transport by extended root systems and soil-water diffusion. Future research should search for mechanisms that explain how the different hypotheses can generate the patterns observed here and test the ability of self-organization to match the birth- and death dynamics of fairy circles and their regional patterns in the density and size with respect to environmental gradients.