  Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron (who
treated the case $e=14$), we prove that the moduli space of polarized K3
surfaces of degree $2e$ contains complete curves for all $e\geq 62$ and for
some sporadic lower values of $e$ (starting at $14$). We also construct
complete curves in the moduli spaces of polarized hyper-K\"ahler manifolds of
$\mathrm{K3}^{[n]}$-type or $\mathrm{Kum}_n$-type for all $n\ge 1$ and
polarizations of various degrees and divisibilities.
