Abstract. The criterion of envy-freeness, according to which no agent should prefer any of his neighbours'allocation to his own, has become a central part of the economic theory of distributive justice. It essentially corresponds to the need to express an ideal of equality in societies where preferences and endowments are heterogeneous. This paper surveys various formulations of the idea of envy-freeness, starting with the simple distribution model, then adding the possibility for agents to have different native talent endowments, and finally moving to models with production. Many classical results are displayed, but emphasis is also put on recent developments, mainly the ideas of 'minimizing envy'and of 'absence of domination'. Five major difficulties facing envyfreeness are identified and listed as directions of future research.