Monte Carlo simulations are used to map out the complete phase diagram of hard body UFO systems, in which the particles are composed of a concentric sphere and thin disk. The equation of state and phase behavior are determined for a range of relative sizes of the sphere and disk. We show that for relatively large disks, nematic and solid phases are observed in addition to the isotropic fluid. For small disks, two different solid phases exist. For intermediate sizes, only a disordered fluid phase is observed. The positional and orientational structure of the various phases are examined. We also compare the equations of state and the nematic-isotropic coexistence densities with those predicted by an extended Onsager theory using virial coefficients up to B(8).