The need for safe, therapeutically effective, and patient-compliant drug delivery systems continuously leads researchers to design novel tools and strategies. Clay minerals are widely used in drug products both as excipients and active agents but, in recent years, there has been a growing interest in research aimed at the development of new organic or inorganic nanocomposites. The attention of the scientific community has been drawn by nanoclays, thanks to their natural origin, worldwide abundance, availability, sustainability, and biocompatibility. In this review, we focused our attention on the studies inherent to the pharmaceutical and biomedical applications of halloysite and sepiolite, and their semi-synthetic or synthetic derivatives, as drug delivery systems. After having described the structure of both materials and their biocompatibility, we delineate the use of the nanoclays to enhance the stability, the controlled release, the bioavailability, and the adsorption properties of drugs. Several types of surface functionalization have been discussed, showing that these materials could be used for the development of an innovative therapeutic approach.