Antimicrobial resistance (AMR) is a growing global crisis with an increasing number of untreatable or exceedingly difficult-to-treat bacterial infections, due to their growing resistance to existing drugs. It is predicted that AMR will be the leading cause of death by 2050. In addition to ongoing efforts on preventive strategies and infection control, there is ongoing research towards the development of novel vaccines, antimicrobial agents, and optimised diagnostic practices to address AMR. However, developing new therapeutic agents and medicines can be a lengthy process. Therefore, there is a parallel ongoing worldwide effort to develop materials for optimised drug delivery to improve efficacy and minimise AMR. Examples of such materials include functionalisation of surfaces so that they can become self-disinfecting or non-fouling, and the development of nanoparticles with promising antimicrobial properties attributed to their ability to damage numerous essential components of pathogens. A relatively new class of materials, metal–organic frameworks (MOFs), is also being investigated for their ability to act as carriers of antimicrobial agents, because of their ultrahigh porosity and modular structures, which can be engineered to control the delivery mechanism of loaded drugs. Biodegradable polymers have also been found to show promising applications as antimicrobial carriers; and, recently, several studies have been reported on delivery of antimicrobial drugs using composites of MOF and biodegradable polymers. This review article reflects on MOFs and polymer–MOF composites, as carriers and delivery agents of antimicrobial drugs, that have been studied recently, and provides an overview of the state of the art in this highly topical area of research.