The great increase in the production and consumption of plastics has resulted in large amounts of plastic wastes, creating a serious problem in terms of their environmentally friendly disposal. The need for the production of more environmentally friendly polymers gave birth to the production of biodegradable, and more recently, biobased polymers, used in the production of biodegradable or biobased plastics. Although the percentage of currently produced bioplastics is rather small, almost 1% compared to petrochemical-based plastics, inevitably is going to significantly increase in the near future due to strict legislation recently posed by the European Union and other countries’ Governments. Thus, recycling strategies that have been developed could be disturbed and the economic balance of this sector could be destabilized. In the present review, the recycling of the polymer mainly used in food plastic packaging, i.e., poly(ethylene terephthalate), PET is examined together with its counterparts from the biobased polymers, i.e., poly(lactic acid), PLA (already replacing PET in several applications), poly(3-hydroxybutyrate), PHB and poly(ethylene furanoate), PEF. Methods for the chemical recycling of these materials together with the chemical products obtained are critically reviewed. Specifically, hydrolysis, alcoholysis and glycolysis. Hydrolysis (i.e., the reaction with water) under different environments (alkaline, acidic, neutral), experimental conditions and catalysts results directly in the production of the corresponding monomers, which however, should be separated in order to be re-used for the re-production of the respective polymer. Reaction conditions need to be optimized with a view to depolymerize only a specific polymer, while the others remain intact. Alcoholysis (i.e., the reaction with some alcohol, methanol or ethanol) results in methyl or ethyl esters or diesters that again could be used for the re-production of the specific polymer or as a source for producing other materials. Glycolysis (reaction with some glycol, such as ethylene, or diethylene glycol) is much studied for PET, whereas less studied for the biopolymers and seems to be a very promising technique. Oligomers having two terminal hydroxyl groups are produced that can be further utilized as starting materials for other value-added products, such as unsaturated polyester resins, methacrylated crosslinked resins, biodegradable polyurethanes, etc. These diols derived from both PET and the bio-based polymers can be used simultaneously without the need for an additional separation step, in the synthesis of final products incorporating biodegradable units in their chemical structure.