  We discuss the cosmic history of supernovae on the basis of various
assumptions and recent data on the star formation history.
  We show that supernova rates as a function of redshift can be used to place
significant constraints on progenitor models, on the star formation history,
and on the importance of dust obscuration.
  We demonstrate that it is unlikely that the current observational indications
for the existence of a cosmological constant are merely an artifact of the
dominance of different progenitor classes at different redshift intervals.
