<p>Non-Orthogonal Multiple Access (NOMA), which has been known to achieve outstanding spectral efficiency and massive connectivity, has been identified as an intriguing multiple-access solution for 5G and beyond wireless networks. Power allocation strategies are of crucial importance among the other factors that influence the performance of NOMA, as they determine the success of Successive Interference Cancellation (SIC) at the receiver. This review presents a synthesis of recent developments in the field of power allocation algorithms in downlink NOMA systems and, especially, algorithms that improve SIC robustness. Some of the major techniques that have been discussed include the dynamic adjustment of power coefficients based on user pairing, real-time channel state information (CSI), and quality of service (QoS) requirements. The review points to comparative results of the available research, indicating that sophisticated optimization techniques, and in many cases, machine learning, can provide better throughput, outage probability, and fairness than fixed and proportional allocation schemes. Lastly, the paper points out existing research gaps and future research directions in the development of flexible and interference-resistant power allocation schemes for next-generation high-capacity and low-latency communications networks.</p>