All animals studied to date are associated with symbiotic communities of microorganisms. These animal microbiotas often play important roles in normal physiological function and susceptibility to disease; predicting their responses to perturbation represents an essential challenge for microbiology. Most studies of microbiome dynamics test for patterns in which perturbation shifts animal microbiomes from a healthy to a dysbiotic stable state. Here, we consider a complementary alternative: that the microbiological changes induced by many perturbations are stochastic, and therefore lead to transitions from stable to unstable community states. The result is an 'Anna Karenina principle' for animal microbiomes, in which dysbiotic individuals vary more in microbial community composition than healthy individuals-paralleling Leo Tolstoy's dictum that "all happy families look alike; each unhappy family is unhappy in its own way". We argue that Anna Karenina effects are a common and important response of animal microbiomes to stressors that reduce the ability of the host or its microbiome to regulate community composition. Patterns consistent with Anna Karenina effects have been found in systems ranging from the surface of threatened corals exposed to above-average temperatures, to the lungs of patients suffering from HIV/AIDs. However, despite their apparent ubiquity, these patterns are easily missed or discarded by some common workflows, and therefore probably underreported. Now that a substantial body of research has established the existence of these patterns in diverse systems, rigorous testing, intensive time-series datasets and improved stochastic modelling will help to explore their importance for topics ranging from personalized medicine to theories of the evolution of host-microorganism symbioses.