  We discuss phenomena arising from the combined influence of electron
correlation and spin-orbit coupling, with an emphasis on emergent quantum
phases and transitions in heavy transition metal compounds with 4d and 5d
elements. A common theme is the influence of spin-orbital entanglement produced
by spin-orbit coupling, which influences the electronic and magnetic structure.
In the weak-to-intermediate correlation regime, we show how non-trivial
band-like topology leads to a plethora of phases related to topological
insulators. We expound these ideas using the example of pyrochlore iridates,
showing how many novel phases such as the Weyl semi-metal, axion insulator,
topological Mott insulator, and topological insulators may arise in this
context. In the strong correlation regime, we argue that spin-orbital
entanglement fully or partially removes orbital degeneracy, reducing or
avoiding the normally ubiquitous Jahn-Teller effect. As we illustrate for the
honeycomb lattice iridates and double perovskites, this leads to enhanced
quantum fluctuations of the spin-orbital entangled states and the chance to
promote exotic quantum spin liquid and multipolar ordered ground states.
Connections to experiments, materials, and future directions are discussed.
