Polymers are essential materials in our daily life. The synthesis of value-added polymers is mainly performed from fossil fuel-derived monomers. However, the adoption of the circular economy model based on the bioeconomy will reduce the dependence on fossil fuels. In this context, biorefineries have emerged to convert biomass into bioenergy and produce high value-added products, including molecules that can be further used as building blocks for the synthesis of biopolymers and bioplastics. The achievement of catalytic systems able to polymerize the natural monomer counterparts, such as terpenes or terpenoids, is still a challenge in the development of polymers with good mechanical, thermal, and chemical properties. This review describes the most common types of bioplastics and biopolymers and focuses specifically on the polymerization of terpenes and terpenoids, which represent a source of promising monomers to create bio-based polymers and copolymers.