The first sentence of Leo Tolstoy's novel Anna Karenina is: "Happy families are all alike; every unhappy family is unhappy in its own way." Here Tolstoy means that for a family to be happy, several key aspects must be given (such as good health of all family members, acceptable financial security, and mutual affection). If there is a deficiency in any one or more of these key aspects, the family will be unhappy. In this paper we introduce the Anna Karenina principle as a concept that can explain success in science. Here we will refer to three central areas in modern science in which scarce resources will most usually lead to failure: (1) peer review of research grant proposals and manuscripts (money and journal space as scarce resources), (2) citation of publications (reception as a scarce resource), and (3) new scientific discoveries (recognition as a scarce resource). If resources are scarce (journal space, funds, reception, and recognition), there can be success only when several key prerequisites for the allocation of the resources are fulfilled. If any one of these prerequisites is not fulfilled, the grant proposal, manuscript submission, the published paper, or the discovery will not be successful.