Chronic wounds are a major concern for global health, affecting millions of individuals worldwide. As their occurrence is correlated with age and age-related comorbidities, their incidence in the population is set to increase in the forthcoming years. This burden is further worsened by the rise of antimicrobial resistance (AMR), which causes wound infections that are increasingly hard to treat with current antibiotics. Antimicrobial bionanocomposites are an emerging class of materials that combine the biocompatibility and tissue-mimicking properties of biomacromolecules with the antimicrobial activity of metal or metal oxide nanoparticles. Among these nanostructured agents, zinc oxide (ZnO) is one of the most promising for its microbicidal effects and its anti-inflammatory properties, and as a source of essential zinc ions. This review analyses the most recent developments in the field of nano-ZnO–bionanocomposite (nZnO-BNC) materials—mainly in the form of films, but also hydrogel or electrospun bandages—from the different preparation techniques to their properties and antibacterial and wound-healing performances. The effect of nanostructured ZnO on the mechanical, water and gas barrier, swelling, optical, thermal, water affinity, and drug-release properties are examined and linked to the preparation methods. Antimicrobial assays over a wide range of bacterial strains are extensively surveyed, and wound-healing studies are finally considered to provide a comprehensive assessment framework. While early results are promising, a systematic and standardised testing procedure for the comparison of antibacterial properties is still lacking, partly because of a not-yet fully understood antimicrobial mechanism. This work, therefore, allowed, on one hand, the determination of the best strategies for the design, engineering, and application of n-ZnO-BNC, and, on the other hand, the identification of the current challenges and opportunities for future research.