Recent observations1 indicate the existence of significant largescale structure on scales >40 ho-1 Mpc. That this is also the typical scale for surviving structure in the hot dark matter models for galaxy formation (for example, neutrinos with masses mv ~ 30 eV) may not be a coincidence (ref. 2; D. N. Schramm, Fermilab preprint 86/70, 1986). If neutrinos are to be reinstated as the leading candidates for the dark matter of the Universe, it becomes crucial to understand the origin of their mass in the context of the superstring3-9. Here we examine the possibility for small but non-zero neutrino masses in a class of superstring models 6 giving rise to acceptable low-energy phenomenology.