The pyrochlore oxides A 2 B 2 O 7 exhibit a complex interplay between geometrical frustration, electronic correlations, and spin-orbit coupling, 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 manybody 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 spin-orbit coupling and local electron correlations are treated within the LDA+U and LDA+dynamical meanfield theory formalisms. We also discuss the technical aspects of these calculations. arXiv:1810.09596v2 [cond-mat.str-el]