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Molecular dynamics simulations were performed to examine different aspects of radiation damage in materials having the bixbyite and NaCl structures. The displacement threshold energy, Ed, and the Frenkel pair formation energy, Efp, were investigated. The Ed values show a significant dependence on the Primary Knock-on Atom (PKA) direction and some correlation between Ed values in the different materials can be observed. No functional relation between Ed and Efp for materials with the same structures was found other than the obvious one that a higher value of Ed corresponds to a higher Efp. Low and high energy cascades in Er2O3 were performed. The Er PKA, which is much heavier than the O PKA, produces more sub-cascades which spread over a shorter distance. The resulting collisional damage was investigated and analysed. Temperature accelerated dynamics and kinetic Monte Carlo methods were used to examine diffusion of point defects in Er2O3. Some of the transitions were found to be rank 2 saddles and a strategy for determining the exponential prefactors was introduced for these cases. The most mobile defect is the oxygen vacancy with lowest energy transition barrier for diffusion of 1 eV. Other defects have very high energy barriers but the energy barrier for recombination between the vacancy and interstitial is found to be dramatically reduced for closely separated Er defects. Radiation damage was examined close to twist grain boundaries in MgO and in nanocrystalline MgO. Grain boundaries were shown to be sinks for interstitial defects with a build up of vacancies in the crystal grains away from the boundaries. The transition barriers for both defects to a grain boundary showed a reduction within 3 layers from the boundary. However there is no net reduction in system energy for a vacancy in the grain boundary compared to a vacancy in the bulk whereas a significant energy reduction is observed for the interstitial in the grain boundary compared with the bulk. These energies show the possible of vacancies to be accumulated in the crystal after collision cascades while interstitials are trapped on the grain boundary.