Neutral models of evolution assume the absence of natural selection. Formerly confined to ecology and evolutionary biology, neutral models are spreading. In recent years they’ve been applied to explaining the diversity of baby names, scientific citations, cryptocurrencies, pot decorations, literary lexica, tumour variants and much more besides. Here, we survey important neutral models and highlight their similarities. We investigate the most widely used tests of neutrality, show that they are weak and suggest more powerful methods. We conclude by discussing the role of neutral models in the explanation of diversity. We suggest that the ability of neutral models to fit low-information distributions should not be taken as evidence for the absence of selection. Nevertheless, many studies, in increasingly diverse fields, make just such claims. We call this tendency ‘neutral syndrome’.