  The unrivalled, extreme luminosities of gamma-ray bursts (GRBs) make them the
favored beacons for sampling the high redshift Universe. To employ GRBs to
study the cosmic terrain -- e.g., star and galaxy formation history -- GRB
luminosities must be calibrated, and the luminosity function versus redshift
must be measured or inferred. Several nascent relationships between gamma-ray
temporal or spectral indicators and luminosity or total energy have been
reported. These measures promise to further our understanding of GRBs once the
connections between the luminosity indicators and GRB jets and emission
mechanisms are better elucidated. The current distribution of 33 redshifts
determined from host galaxies and afterglows peaks near z ~ 1, whereas for the
full BATSE sample of long bursts, the lag-luminosity relation predicts a broad
peak z ~ 1-4 with a tail to z ~ 20, in rough agreement with theoretical models
based on star formation considerations. For some GRB subclasses and apparently
related phenomena -- short bursts, long-lag bursts, and X-ray flashes -- the
present information on their redshift distributions is sparse or entirely
lacking, and progress is expected in Swift era when prompt alerts become
numerous.
