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We study the problem of distributing a set of indivisible goods among agents with additive valuations in a fair manner. The fairness notion under consideration is envy-freeness up to any good (EFX). Despite significant efforts by many researchers for several years, the existence of EFX allocations has not been settled beyond the simple case of two agents. In this article, we show constructively that an EFX allocation always exists for three agents. Furthermore, we falsify the conjecture of Caragiannis et al. by showing an instance with three agents for which there is a partial EFX allocation (some goods are not allocated) with higher Nash welfare than that of any complete EFX allocation. [ABSTRACT FROM AUTHOR]