Most diseases that affect human beings across the world are now treated with drugs of organic origin. However, some of these are associated with side effects, toxicity, and resistance phenomena. For the treatment of many illnesses, the development of new molecules with pharmacological potential is now an urgent matter. The biological activities of metal complexes have been reported to have antitumor, antimicrobial, anti-inflammatory, anti-infective and antiparasitic effects, amongst others. Metal complexes are effective because they possess unique properties. For example, the complex entity possesses the effective biological activity, then the formation of coordination bonds between the metal ions and ligands is controlled, metal ions provide it with extraordinary mechanisms of action because of characteristics such as d-orbitals, oxidation states, and specific orientations; metal complexes also exhibit good stability and good physicochemical properties such as water solubility. Platinum is a transition metal widely used in the design of drugs with antineoplastic activities; however, platinum is associated with side effects which have made it necessary to search for, and design, novel complexes based on other metals. Copper is a biometal which is found in living systems; it is now used in the design of metal complexes with biological activities that have demonstrated antitumoral, antimicrobial and anti-inflammatory effects, amongst others. In this review, we consider the open horizons of Cu(II)- and Pt(II)-based complexes, new trends in their design, their synthesis, their biological activities and their targets of action.