Plastics are very useful materials and present numerous advantages in the daily life of individuals and society. However, plastics are accumulating in the environment and due to their low biodegradability rate, this problem will persist for centuries. Until recently, oceans were treated as places to dispose of litter, thus the persistent substances are causing serious pollution issues. Plastic and microplastic waste has a negative environmental, social, and economic impact, e.g., causing injury/death to marine organisms and entering the food chain, which leads to health problems. The development of solutions and methods to mitigate marine (micro)plastic pollution is in high demand. There is a knowledge gap in this field, reason why research on this thematic is increasing. Recent studies reported the biodegradation of some types of polymers using different bacteria, biofilm forming bacteria, bacterial consortia, and fungi. Biodegradation is influenced by several factors, from the type of microorganism to the type of polymers, their physicochemical properties, and the environment conditions (e.g., temperature, pH, UV radiation). Currently, green environmentally friendly alternatives to plastic made from renewable feedstocks are starting to enter the market. This review covers the period from 1964 to April 2020 and comprehensively gathers investigation on marine plastic and microplastic pollution, negative consequences of plastic use, and bioplastic production. It lists the most useful methods for plastic degradation and recycling valorization, including degradation mediated by microorganisms (biodegradation) and the methods used to detect and analyze the biodegradation.