Massive Thorne-Zytkow objects (TZOs) are red supergiants with neutron cores which are expected to form as a result of unstable mass transfer in massive X-ray binaries, or the direct collision of a neutron star with a massive companion after a supernova. We estimate a total TZO birthrate in the Galaxy of ~2x10^-4 yr^-1. Thus, for a characteristic TZO lifetime of 10^5-10^6 yr, there should be 20-200 TZOs in the Galaxy at present. These can be distinguished from ordinary red supergiants because of anomalously high surface abundances of lithium and rp-process elements, produced in the TZO interior. The TZO phase ends when either the star has exhausted its rp-process seed elements or the envelope mass decreases below a critical mass (~14 M_solar). We confirm earlier conjectures that nuclear burning then becomes inefficient and a neutrino runaway ensues; this can lead to dynamical accretion of matter near the core on to the neutron star and its spin-up to spin periods as low as ~10 ms. The fate of the massive envelope is not entirely clear. If a significant fraction can be accreted on to the core, the formation of a black hole becomes likely. Part of the envelope may collapse into a massive disc, which may ultimately become gravitationally unstable and lead to the formation of planets or even low-mass stars. We discuss the various possible outcomes, and suggest a possible link between massive TZOs and soft X-ray transients.