  We review the current state of results about the half-wave maps equation on
the domain $\mathbb{R}^d$ with target $\mathbb{S}^2$. In particular, we focus
on the energy-critical case $d=1$, where we discuss the classification of
traveling solitary waves and a Lax pair structure together with its
implications (e.\,g.~invariance of rational solutions and infinitely many
conservation laws on a scale of homogeneous Besov spaces). Furthermore, we also
comment on the one-dimensional space-periodic case. Finally, we list some open
problem for future research.
