Studies on bionanocomposite drug carriers are a key area in the field of active substance delivery, introducing innovative approaches to improve drug therapy. Such drug carriers play a crucial role in enhancing the bioavailability of active substances, affecting therapy efficiency and precision. The targeted delivery of drugs to the targeted sites of action and minimization of toxicity to the body is becoming possible through the use of these advanced carriers. Recent research has focused on bionanocomposite structures based on biopolymers, including lipids, polysaccharides, and proteins. This review paper is focused on the description of lipid-containing nanocomposite carriers (including liposomes, lipid emulsions, lipid nanoparticles, solid lipid nanoparticles, and nanostructured lipid carriers), polysaccharide-containing nanocomposite carriers (including alginate and cellulose), and protein-containing nanocomposite carriers (e.g., gelatin and albumin). It was demonstrated in many investigations that such carriers show the ability to load therapeutic substances efficiently and precisely control drug release. They also demonstrated desirable biocompatibility, which is a promising sign for their potential application in drug therapy. The development of bionanocomposite drug carriers indicates a novel approach to improving drug delivery processes, which has the potential to contribute to significant advances in the field of pharmacology, improving therapeutic efficacy while minimizing side effects.