Electronic theory calculations are applied to the study of the UO molecule and the UO+ ion. Relativistic effective core potentials are used along with the accompanying valence spinorbit operators. Polarized double-zeta and triple-zeta basis sets are used. Molecular orbitals are obtained from state-averaged multiconfiguration self-consistent field calculations and then used in multireference spin-orbit configuration interaction calculations with a number of millions of terms. The ground state of UO has open shells of 5f(3)7s(1), angular momentum Omega = 4, and a spin-orbit-induced avoided crossing near the equilibrium internuclear distance. Many UO excited states are studied with rotational constants, intensities, and experimental comparisons. The ground state of UO+ is of 5f(3) nature with Omega = 9/2. Many UO+ excited states are also studied. The open-shell nature of both UO and UO+ leads to many low-lying excited states.