This review discusses the combined effects of electron correlation and spin-orbit coupling in heavy transition metal compounds with 4d and 5d elements, focusing on emergent quantum phases and transitions. A key theme is the influence of spin-orbit entanglement, which affects electronic and magnetic structures. In the weak-to-intermediate correlation regime, non-trivial band-like topology leads to phases such as topological insulators, Weyl semi-metals, and axion insulators. In the strong correlation regime, spin-orbit entanglement reduces orbital degeneracy, mitigating the Jahn-Teller effect and enabling exotic quantum spin liquid and multipolar ordered states. The review highlights experimental and theoretical studies of pyrochlore iridates, honeycomb iridates, and double perovskites, discussing their electronic structures, magnetic properties, and phase transitions. It also explores the role of many-body effects, interactions with rare earth moments, and the implications of strong spin-orbit coupling for quantum phases. The review concludes with a discussion of open issues and future directions in this field.This review discusses the combined effects of electron correlation and spin-orbit coupling in heavy transition metal compounds with 4d and 5d elements, focusing on emergent quantum phases and transitions. A key theme is the influence of spin-orbit entanglement, which affects electronic and magnetic structures. In the weak-to-intermediate correlation regime, non-trivial band-like topology leads to phases such as topological insulators, Weyl semi-metals, and axion insulators. In the strong correlation regime, spin-orbit entanglement reduces orbital degeneracy, mitigating the Jahn-Teller effect and enabling exotic quantum spin liquid and multipolar ordered states. The review highlights experimental and theoretical studies of pyrochlore iridates, honeycomb iridates, and double perovskites, discussing their electronic structures, magnetic properties, and phase transitions. It also explores the role of many-body effects, interactions with rare earth moments, and the implications of strong spin-orbit coupling for quantum phases. The review concludes with a discussion of open issues and future directions in this field.