Mergers of binary neutron star systems are among the most fascinating astrophysical sites in our universe. Computational simulations of these events require modeling of the hydrodynamics of nuclear matter at extreme densities and temperatures, strong dynamical gravitational fields, neutrino-matter interactions, and more. Moreover, they are computationally costly requiring the use of highly performant simulation software. In this report, we summarize our recent results in exploring the impact of the nuclear equation of state on the dynamics and observable features of binary neutron star mergers. To this extent, we systematically vary select nuclear matter properties and explore the results in numerical general relativistic hydrodynamical simulations. Our results show that some details of the merger, such as the frequencies of the post-merger gravitational-wave signal and the mass of the shock-heated ejecta, are sensitive to the EOS close to saturation density while others, such as the contraction of the remnant neutron star and the mass of the tidal ejecta, are sensitive to the behavior at higher densities. Furthermore, we show preliminary results of our simulation setup to explore the long-term evolution of the post-merger phase in axisymmetry.