The pyrochlore oxides [Formula: see text]O7 exhibit a complex interplay between geometrical frustration, electronic correlations, and spin-orbit coupling (SOC), due to the lattice structure and active charge, spin, and orbital degrees of freedom. Understanding the properties of these materials is a theoretical challenge, because their intricate nature depends on material-specific details and quantum many-body effects. Here we review our recent studies based on first-principles calculations and quantum many-body theories for 4d and 5d pyrochlore oxides with B = Mo, Os, and Ir. In these studies, the SOC and local electron correlations are treated within the local density approximation (LDA) + U and LDA + dynamical mean-field theory formalisms. We also discuss the technical aspects of these calculations.