Experimental results and simulations of imploding fusion concepts have identified the Rayleigh-Taylor (RT) instability as one of the largest inhibitors to achieving fusion. Understanding the origin and development of the RT instability will allow for the development of mitigating measures to dampen the instability growth, thus improving the chance that fusion concepts such as inertial confinement fusion (ICF) are successful. A study of 1D and 2D simulations are presented for investigating RT instability growth in deceleration stage of imploding geometries. Two cases of laser-driven implosion geometry, Cartesian and cylindrical, are used to study late stage deceleration-phase RT instability development on the interior surface of imploding targets. FLASH's hydrodynamic (HD) and magnetohydrodynamic (MHD) modeling capabilities are used for different laser and target parameters in order to study the RT instability and the impact of externally applied magnetic fields on their evolution. Several simulation regimes have been identified that provide novel insight into the impact that a seeded magnetic field can have on RT instability growth and the conditions under which magnetic field stabilization of the RT instability is observable. Finally, future work and recommendations are made. / Master of Science / The direction for the future of renewable energy is uncertain at this time; however, it is known that the future of human energy consumption must be green in order to be sustainable. Fusion energy presents an opportunity for an unlimited clean renewable energy source that has yet to be realized. Fusion is achieved only by overcoming the earthly limitations presented by trying to replicate conditions at the interior of stellar structures. The pressures, temperature, and densities seen in the interior of stars are not easily reproduced, and thus human technology must be developed to reach these difficult stellar conditions in order to harvest fusion energy. There are two main branches of developmental technology geared towards achieving the difficult conditions controlled nuclear fusion presents, magnetic confinement fusion (MCF) and inertial confinement fusion (ICF)[17]. Yet in both approaches barriers exist which have thwarted the efforts toward reaching fusion ignition which must be addressed through scientific discovery. Successfully reaching ignition is only the first step in the ultimate pursuit of a self sustaining fusion reactor. This work will focus on the experimental ICF configuration, and on one such inhibitor toward achieving ignition, the Rayleigh-Taylor (RT) instability. The RT instability develops on the surfaces of the fusion fuel capsules, targets, and causes nonuniform compression of the target. This nonuniform compression of the target leads to lower pressures and densities through the material mixing of fusion fuel and the capsule shell, which ultimately leads to challenges with reaching fusion ignition. The work presented here was performed utilizing the University of Chicago's FLASH code, which is a state-of-the-art open source radiation magneto-hydrodynamic (MHD) code used for plasma and astrophysics computational modeling [11]. Simulations of the RT instability are performed using FLASH in planar and cylindrical geometries to explore fundamental Rayleigh-Taylor instability evolution for these two different geometries. These geometries provide easier access for experimental diagnostics to probe RT dynamics. Additionally, the impact of externally applied magnetic fields are explored in an effort to examine if and how the detrimental instability can be controlled.