Antimicrobial resistance (AMR) is a significant global threat to both public health and the environment. The emergence and expansion of AMR is sustained by the enormous diversity and mobility of antimicrobial resistance genes (ARGs). Different mechanisms of horizontal gene transfer (HGT), including conjugation, transduction, and transformation, have facilitated the accumulation and dissemination of ARGs in Gram-negative and Gram-positive bacteria. This has resulted in the development of multidrug resistance in some bacteria. The most clinically significant ARGs are usually located on different mobile genetic elements (MGEs) that can move intracellularly (between the bacterial chromosome and plasmids) or intercellularly (within the same species or between different species or genera). Resistance plasmids play a central role both in HGT and as support elements for other MGEs, in which ARGs are assembled by transposition and recombination mechanisms. Considering the crucial role of MGEs in the acquisition and transmission of ARGs, a potential strategy to control AMR is to eliminate MGEs. This review discusses current progress on the development of chemical and biological approaches for the elimination of ARG carriers.