This thesis studies superradiance and subradiance in the normal modes of excitation of systems of resonant nuclei. The nuclei are treated as classical electric dipole oscillators interacting through their radiation fields. This coupling radically alters the radiative decay of the system relative to the decay of an isolated nucleus, and leads to both superradiant and subradiant states with strongly enhanced or suppressed radiative decay rates, and with normal mode frequencies which can be strongly shifted from the natural resonance frequency. This issue is of considerable current interest because it is now possible to create spatially coherent single exciton nuclear states by illuminating crystals containing resonant Mossbauer nuclei with synchrotron radiation pulses. The purpose of this thesis is to solve for the radiative eigenmodes in a system of nuclei and to investigate superradiance and subradiance both in the eigenmode solutions and in the synchrotron radiation produced nuclear exciton state.