  Emergence of novel quantum ground states in correlated electron systems with
strong spin-orbit coupling has been a recent subject of intensive studies.
While it has been realized that spin-orbit coupling can provide non-trivial
band topology in weakly interacting electron systems, as in topological
insulators and semi-metals, the role of electron-electron interaction in
strongly spin-orbit coupled systems has not been fully understood. The
availability of new materials with significant electron correlation and strong
spin-orbit coupling now makes such investigations possible. Many of these
materials contain 5d or 4d transition metal elements; the prominent examples
are iridium oxides or iridates. In this review, we succinctly discuss recent
theoretical and experimental progress on this subject. After providing a brief
overview, we focus on pyrochlore iridates and three-dimensional honeycomb
iridates. In pyrochlore iridates, we discuss the quantum criticality of the
bulk and surface states, and the relevance of the surface/boundary states in a
number of topological and magnetic ground states, both in the bulk and thin
film configurations. Experimental signatures of these boundary and bulk states
are discussed. Domain wall formation and strongly-direction-dependent
magneto-transport are also discussed. Regarding the three-dimensional honeycomb
iridates, we consider possible quantum spin liquid phases and unusual magnetic
orders in theoretical models with strongly bond-dependent interactions. These
theoretical ideas and results are discussed in light of recent resonant X-ray
scattering experiments on three-dimensional honeycomb iridates. We also
contrast these results with the situation in two-dimensional honeycomb
iridates. We conclude with the outlook on other related systems.
