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Non-orthogonal multiple access (NOMA) has been recognized as a promising multi-user access technique for the next generation cellular communication networks. In this chapter, we first review the basic concepts of downlink NOMA transmission and introduce the two-user, multi-user, and multi-channel NOMA schemes. Then, we investigate the optimal power allocation for these downlink NOMA schemes under different performance criteria, including the maximin fairness, sum rate, and energy efficiency. User weights and quality-of-service (QoS) constraints are taken into account. We show that in most cases the optimal NOMA power allocation can be analytically characterized, while in other cases the NOMA power allocation problems can be numerically solved via convex optimization methods.