  This paper addresses two aspects concerning the $\theta$-vacuum of Quantum
Chromodynamics. First, large-$N_c$ chiral perturbation theory is used to
calculate the first two non-trivial cumulants of the distribution of the
winding number, i.\,e. the topological susceptibility, $\chi_\mathrm{top}$, and
the fourth cumulant, $c_4$, up to next-to-leading order. Their large-$N_c$
scaling is discussed, and compared to lattice results. It is found that
$\chi_\mathrm{top}=\mathcal{O}(N_c^0)$, as known before, and
$c_4=\mathcal{O}(N_c^{-3})$, correcting the assumption of
$\mathcal{O}(N_c^{-2})$ in the literature. Second, we discuss the properties of
QCD at $\theta\sim\pi$ using chiral perturbation theory for the case of $2+1$
light flavors, i.\,e. by taking the strange quark mass heavier than the
degenerate up and down quark masses. It is shown that --- in accordance with
previous findings for $N_f=2$ and $N_f=3$ mass-degenerate flavors --- in the
region $\theta\sim\pi$ two vacuum states coexist, which become degenerate at
$\theta=\pi$. The wall tension of the energy barrier between these degenerate
vacua is determined as well as the decay rate of a false vacuum.
