The current global plastics economy is highly linear, with the exceptional performance and low carbon footprint of polymeric materials at odds with dramatic increases in plastic waste. Transitioning to a circular economy that retains plastic in its highest value condition is essential to reduce environmental impacts, promoting reduction, reuse, and recycling. Mechanical recycling is an essential tool in an environmentally and economically sustainable economy of plastics, but current mechanical recycling processes are limited by cost, degradation of mechanical properties, and inconsistent quality products. This review covers the current methods and challenges for the mechanical recycling of the five main packaging plastics: poly(ethylene terephthalate), polyethylene, polypropylene, polystyrene, and poly(vinyl chloride) through the lens of a circular economy. Their reprocessing induced degradation mechanisms are introduced and strategies to improve their recycling are discussed. Additionally, this review briefly examines approaches to improve polymer blending in mixed plastic waste streams and applications of lower quality recyclate.