  Complex oxides with $4d$ and $5d$ transition-metal ions recently emerged as a
new paradigm in correlated electron physics, due to the interplay between
spin-orbit coupling and electron interactions. For $4d$ and $5d$ ions, the
spin-orbit coupling, $\zeta$, can be as large as 0.2-0.4 eV, which is
comparable with and often exceeds other relevant parameters such as Hund's
coupling $J_{\rm H}$, noncubic crystal field splitting $\Delta$, and the
electron hopping amplitude $t$. This gives rise to a variety of
spin-orbit-entangled degrees of freedom and, crucially, non-trivial
interactions between them that depend on the $d$-electron configuration, the
chemical bonding, and the lattice geometry. Exotic electronic phases often
emerge, including spin-orbit assisted Mott insulators, quantum spin liquids,
excitonic magnetism, multipolar orderings and correlated topological
semimetals. This paper provides a selective overview of some of the most
interesting spin-orbit-entangled phases that arise in $4d$ and $5d$
transition-metal compounds.
