Plastics are versatile materials used in a variety of sectors that have seen a rapid increase in their global production. Millions of tonnes of plastic wastes are generated each year, which puts pressure on plastic waste management methods to prevent their accumulation within the environment. Recycling is an attractive disposal method and aids the initiative of a circular plastic economy, but recycling still has challenges to overcome. This review starts with an overview of the current European recycling strategies for solid plastic waste and the challenges faced. Emphasis lies on the recycling of polyolefins (POs) and polyethylene terephthalate (PET) which are found in plastic packaging, as packaging contributes a signification proportion to solid plastic wastes. Both sections, the recycling of POs and PET, discuss the sources of wastes, chemical and mechanical recycling, effects of recycling on the material properties, strategies to improve the performance of recycled POs and PET, and finally the applications of recycled POs and PET. The review concludes with a discussion of the future potential and opportunities of recycled POs and PET.