Many distributive issues involve situations in which initial characteristics make individuals unequal. In view of prevailing moral sentiments, some of these characteristics call for compensating transfers, and some do not. We study the literature on this problem of compensation. This literature follows the distinction between the ethical principle of compensation and that of responsibility. According to the former, a good resource allocation system should neutralize the differential influence over agents' outcomes of the characteristics that elicit compensation. According to the latter, a good resource allocation system should remain neutral with respect to inequality arising from the influence of characteristics that do not elicit compensation. The principle of responsibility can be interpreted as a libertarian principle of natural reward, or as a principle of utilitarian reward. Depending on whether the emphasis is put on the principle of compensation or of responsibility, and depending on how the latter is interpreted, there exist four main families of solutions to compensation problems. We review the axiomatic analyses of these four families of solutions in the different models in which they have been studied. We also review the applications that have been made of these solutions to problems of income taxation, education investment, social mobility and health insurance systems.