We extend Maskin's results on Nash implementation. First, we establish a condition which is both necessary and sufficient for Nash implementability if there are three or more agents (this was the case covered by Maskin's sufficiency result). Second--and more important--we examine the two-agent case (for which there existed no general sufficiency results). The two-agent model is of course the leading case for applications to contracting and bargaining. For this case, too, we establish a condition which is both necessary and sufficient for Nash implementability. We use our theorems to derive simpler sufficiency conditions that are applicable in a wide variety of economic environments.