  We show how Type Ia/II supernova rates at various redshifts can be used to
follow the cosmic star formation rate (CSFR) history. Using the recent e
stimate of the Cosmic Star Formation (CSFR), we compute the predicted cosmic
Type Ia/II supernovae rates.
  We show that while SNII rates provide a direct measurement of the
instantaneous CSFR, the SNIa rate depends on the past history of the CSFR. This
provides us with a new and independent diagnostic of the star formation rate.
The comparison with the data are in good agreement with both local and high-$z$
measurement and provides an additional evid ence of the validity of the CSFR up
to $z = 1$, but suggests higer values at $z > 1$ co nsistent with the presence
of extinction effects. Measurement of SNIa rate up to $z {\s im}$ 1 will be
crucial for our understanding of the star formation history at early epo chs $z
> 1$. We also show that current observations on Type Ia supernovae rate already
give us valubale information on the nature of their progenitors.
