Copper ions bind to biomolecules (e.g., peptides and proteins) playing an essential role in many biological and physiological pathways in the human body. The resulting complexes may contribute to the initiation of neurodegenerative diseases, cancer, and bacterial and viral diseases, or act as therapeutics. Some compounds can chemically damage biological macromolecules and initiate the development of pathogenic states. Conversely, a number of these compounds may have antibacterial, antiviral, and even anticancer properties. One of the most significant current discussions in Cu biochemistry relates to the mechanisms of the positive and negative actions of Cu ions based on the generation of reactive oxygen species, including radicals that can interact with DNA molecules. This review aims to analyze various peptide–copper complexes and the mechanism of their action.