The development of CORMs (CO-releasing molecules) as a prodrug for CO administration in living organisms has attracted significant attention. CORMs offer the promising possibility of a safe and controllable release of CO in low amounts triggered by light, ligands, enzymes, etc. For the targeting of specific tissues or diseases and to prevent possible side effects from metals and other residues after CO release, these CORMs are attached to biocompatible systems, like peptides, polymers, nanoparticles, dendrimers, protein cages, non-wovens, tablets, and metal-organic frameworks. We discuss in this review the known CORM carrier conjugates, in short CORM conjugates, with covalently-bound or incorporated CORMs for medicinal and therapeutic applications. Most conjugates are nontoxic, show increasing half-lives of CO release, and make use of the EPR-effect, but still show problems because of a continuous background of CO release and the absence of an on/off-switch for the CO release.