The increasing consumption of plastics and plastic products results in correspondingly substantial volumes of waste, which poses considerable environmental burdens. With the ongoing environmental actions, the application of circular economy on this waste stream is becoming inevitable. In this paper, the topics of plastics recycling, circular economy on plastics, and challenges to plastic waste recycling are critically reviewed. In the first part of this paper, the development of research on plastic recycling was viewed from 1950 until 2020 using the scientific database Web of Science, and 682 related studies were found and used to assess the changing research priorities along that timeline. The following sections discuss the potentials and requirements to enhance the quality of the produced recycled plastic, in connection with the factors that currently limit it. In conclusion, the quality of recycled plastic is generally determined by the homogeneity of the recovered plastic feed. There are various strategies which could be implemented to overcome the hindrances identified in the paper and to improve the quality of the recycled plastic, such as working on enhanced product designs for minimised waste heterogeneity and controlling the materials’ degree of contamination by applying advanced sorting.