The peach–potato aphid, Myzus persicae (Sulzer), is one of the most important pests of economic crops. It damages the plant directly by consuming nutrients and water and indirectly by transmitting plant viruses. This pest has the unenviable title of having resistance to more insecticides than any other herbivorous insect pest. Due to the development of its resistance to chemical pesticides, it is necessary to find other control options. Consequently, increased efforts worldwide have been undertaken to develop new management approaches for M. persicae. In this review, we highlight the problems associated with the peach–potato aphid, its economic importance, and current management approaches. This review also describes the challenges with current management approaches and their potential solutions, with special focus given to the evolution of insecticidal resistance and sustainable pest management strategies, such as biocontrol agents, entomopathogens, the use of natural plant-derived compounds, and cultural methods. Furthermore, this review provides some successful approaches from the above eco-friendly pest management strategies that show high efficacy against M. persicae.