  We consider nonlinear half-wave equations with focusing power-type
nonlinearity $$ i \pt_t u = \sqrt{-\Delta} \, u - |u|^{p-1} u, \quad \mbox{with
$(t,x) \in \R \times \R^d$} $$ with exponents $1 < p < \infty$ for $d=1$ and $1
< p < (d+1)/(d-1)$ for $d \geq 2$. We study traveling solitary waves of the
form $$ u(t,x) = e^{i\omega t} Q_v(x-vt) $$ with frequency $\omega \in \R$,
velocity $v \in \R^d$, and some finite-energy profile $Q_v \in H^{1/2}(\R^d)$,
$Q_v \not \equiv 0$. We prove that traveling solitary waves for speeds $|v|
\geq 1$ do not exist. Furthermore, we generalize the non-existence result to
the square root Klein--Gordon operator $\sqrt{-\DD+m^2}$ and other
nonlinearities.
  As a second main result, we show that small data scattering fails to hold for
the focusing half-wave equation in any space dimension. The proof is based on
the existence and properties of traveling solitary waves for speeds $|v| < 1$.
Finally, we discuss the energy-critical case when $p=(d+1)/(d-1)$ in dimensions
$d \geq 2$.
