Detailed models of the martian interior must be developed so that the results of future missions to Mars may be interpreted. Ab initio computer simulations are used to investigate the properties of iron sulphides as candidate phases for the martian core. The first simulations of experimentally observed and hypothetical phases of stoichiometric FeS and Fe3S provide details of the properties of these phases at high pressures that have yet to be investigated experimentally. Simulations of liquid iron sulphides constitute the first self-contained theoretical study of these liquids, providing new information about their structural, dynamical and bulk physical properties as a function of pressure, temperature and composition. New models of the martian interior are developed that are geophysically sound, consistent with observations of Mars, incorporate new results from simulations of liquid iron sulphides, and generate temperature, density and seismic wave velocity profiles as a function of depth. Calculated thermal profiles constrain the core to be liquid. For a core compositions of Fe ~16 wt%S and Fe ~27 wt%S the core radius is constrained to ~1500 - 1600 km and ~1600 - 1800 km, respectively. Calculated densities and adiabatic bulk modulus are consistent with a mantle composed of olivines, pyroxenes and garnets, and exclude a perovskite layer at the base of the mantle. New seismic profiles indicate that the average primary seismic wave velocity in the core is only ~3 km s-1. Geology, magnetic, gravity, topography datasets for six regions of Mars are studied. There is no clear trend of correlations between the datasets across the studied regions. Resurfacing is not a cause of first order density and magnetic contrasts. Crustal remanent magnetic anomalies are more likely to be associated with underlying bedrock than with erosional and depositional processes or products. Current understanding of these anomalies is not sufficient to further constrain models of the martian interior
