Cardiovascular diseases are the number one cause of heart failure and death in the world, and the transplantation of the heart is an effective and viable choice for treatment despite presenting many disadvantages (most notably, transplant heart availability). To overcome this problem, cardiac tissue engineering is considered a promising approach by using implantable artificial blood vessels, injectable gels, and cardiac patches (to name a few) made from biodegradable polymers. Biodegradable polymers are classified into two main categories: natural and synthetic polymers. Natural biodegradable polymers have some distinct advantages such as biodegradability, abundant availability, and renewability but have some significant drawbacks such as rapid degradation, insufficient electrical conductivity, immunological reaction, and poor mechanical properties for cardiac tissue engineering. Synthetic biodegradable polymers have some advantages such as strong mechanical properties, controlled structure, great processing flexibility, and usually no immunological concerns; however, they have some drawbacks such as a lack of cell attachment and possible low biocompatibility. Some applications have combined the best of both and exciting new natural/ synthetic composites have been utilized. Recently, the use of nanostructured polymers and polymer nanocomposites has revolutionized the field of cardiac tissue engineering due to their enhanced mechanical, electrical, and surface properties promoting tissue growth. In this review, recent research on the use of biodegradable natural/synthetic nanocomposite polymers in cardiac tissue engineering is presented with forward looking thoughts provided for what is needed for the field to mature.