Building on an idea of Borcherds, Katzarkov, Pantev, and Shepherd-Barron, we prove that the moduli space of polarized K3 surfaces of degree 2e contains complete curves for all and for some sporadic lower values of e (starting at 14). We also construct complete curves in the moduli spaces of polarized hyper-Kähler manifolds of -type or Kum_n-type for all and polarizations of various degrees and divisibilities. This is joint work with Emanuele Macrì.