The number of pet dogs is increasing in many post-industrialised countries, while human fertility is permanently low and individuals have few relatives to rely on or to care for. Most dog owners say their dog is a “family member” or even a “furbaby”. The present review offers a novel theoretical outline to understand why people refer to their dogs as kin. Kinship offers “unconditional love” and the feeling of “being needed” for humans, traits that are now attributed to and highly valued in dogs. The dog keeping runaway theory assumes that due to demographic transition, the uniquely strong, innate preference for social proximity and nurturing infants “ran away” and shifted to pets in the lack of available kin. Western culture reinforces pet dog keeping, and there are no other, e.g., economic constraints, to stop the runaway process. As a result, both humans and dogs might pay a high price. For example, caring for and cuddling pets might decrease the biological fitness of owners by reducing the desire to have children, and it is yet unclear whether owners, in general, gain any mental or physical health advantages compared to non-owners. At the same time, in the case of dogs, artificial selection for infant-like traits such as brachycephalism and caring for a dog as a “surrogate child” can be associated with severe health and behaviour problems. The review also raises the question of whether dogs facilitate or worsen humans’ network complexity, how cultures differ in pet keeping practices, why women are more closely associated with dogs than men, and what other species or artificial agents can be ideal pets.