Antibiotic resistance is one of the greatest threats to global health and food security today. It becomes increasingly difficult to treat infectious disorders because antibiotics, even the newest ones, are becoming less and less effective. One of the ways taken in the Global Plan of Action announced at the World Health Assembly in May 2015 is to ensure the prevention and treatment of infectious diseases. In order to do so, attempts are made to develop new antimicrobial therapeutics, including biomaterials with antibacterial activity, such as polycationic polymers, polypeptides, and polymeric systems, to provide non-antibiotic therapeutic agents, such as selected biologically active nanoparticles and chemical compounds. Another key issue is preventing food from contamination by developing antibacterial packaging materials, particularly based on degradable polymers and biocomposites. This review, in a cross-sectional way, describes the most significant research activities conducted in recent years in the field of the development of polymeric materials and polymer composites with antibacterial properties. We particularly focus on natural polymers, i.e., polysaccharides and polypeptides, which present a mechanism for combating many highly pathogenic microorganisms. We also attempt to use this knowledge to obtain synthetic polymers with similar antibacterial activity.