The possibility of targeted drug delivery to a specific tissue, organ, or cell has opened new promising avenues in treatment development. The technology of targeted delivery aims to create multifunctional carriers that are capable of long circulation in the patient’s organism and possess low toxicity at the same time. The surface of modern synthetic carriers has high structural similarity to the cell membrane, which, when combined with additional modifications, also promotes the transfer of biological properties in order to penetrate physiological barriers effectively. Along with artificial nanocages, further efforts have recently been devoted to research into extracellular vesicles that could serve as natural drug delivery vehicles. This review provides a detailed description of targeted delivery systems that employ lipid and lipid-like nanocages, as well as extracellular vesicles with a high level of biocompatibility, highlighting genetically encoded drug delivery vehicles.