This project is directed toward a transformative improvement in our understanding of materials containing transition-metal ions from the 5d block of the periodic table. These elements are distinguished by a strong spin-orbit effect that tends to twist the spin of the electrons as they orbit around the nucleus, endowing these materials with useful or unusual magnetic, optical, and electronic properties. The project is built upon a materials discovery paradigm in which first-principles computational methods will be used to scan through candidate materials, identifying promising candidates for directed synthesis and in-depth experimental study. The immediate goal is to improve our scientific understanding of the competing interactions that give rise to functional behavior in this class of materials. Longer-term goals include achieving large magneto-crystalline anisotropy in crystals with mixed light and heavy transition-metal ions, finding new topological insulators, demonstrating unusual superconducting pairing mechanisms or topological superconductivity, and developing materials with giant magnetoelectric, multiferroic, or magneto-optic effects.