  QSOs allow study of the evolution of the relationship between black holes in
galactic nuclei and their host galaxies. The black hole mass can be derived
from the widths of the broad emission lines, and the stellar velocity
dispersion (sigma_*) of the host galaxy can be inferred from the narrow
emission lines. Results based on [OIII] and [OII] line widths indicate that the
black hole mass - sigma_* relationship, at redshifts up to z ~ 2, is consistent
with no evolution or an increase of up to ~ 0.5 dex in black hole mass at fixed
sigma_*. CO line widths offer an estimate of sigma_* for luminous QSOs at high
redshifts. The available objects from z ~ 4 to 6 have very massive black holes,
\~ 10^9.5 M_sun, but their CO line widths suggest much smaller host galaxies
than would be expected by the local black hole mass - sigma_* relationship. The
most massive black holes must continue to reside in comparatively modest
galaxies today, because their number density inferred from QSO statistics
exceeds the present-day abundance of proportionally massive galaxies.
