Abstract: The idea of using a directional time-varying(rotating) driving magnetic field (the alternant theta-Z-pinch configuration) or the multi-nested theta-Z-pinch configuration to suppress magneto-Rayleigh-Taylor instabilities in a dynamic Z-pinch was proposed between 2014 and 2016. This paper reports the latest research progress on magneto-Rayleigh-Taylor instabilities in alternant/nested theta-Z-pinches. The magneto-Rayleigh-Taylor instability growth of a properly optimized alternant/nested theta-Z-pinch is significantly lower than that of a pure theta-pinch or Z-pinch. The instability is completely stabilized at a certain thickness of the liner. These show the bright potential of alternant/nested theta-Z-pinch for theta-Z-pinch liner inertial fusion.