Type-Ia supernova rates and the progenitor problem, a review
  The identity of the progenitor systems of type-Ia supernovae (SNe Ia) is a
major unsolved problem in astrophysics. SN Ia rates are providing some striking
clues. We review the basics of SN rate measurement, preach about some sins of
SN rate measurement and analysis, and illustrate one of these sins with an
analogy about Martian scientists. We review the recent progress in measuring SN
Ia rates in various environments and redshifts, and their use to reconstruct
the SN Ia delay time distribution (DTD) -- the SN rate versus time that would
follow a hypothetical brief burst of star formation. A good number of DTD
measurements, using a variety of methods, appear to be converging. At delays
1<t<10 Gyr, these measurements show a similar, ~t^-1, power-law shape. The DTD
peaks at the shortest delays probed, although there is still some uncertainty
regarding its precise shape at t<1 Gyr. At face value, this result supports the
idea of a double-degenerate progenitor origin for SNe Ia. Single-degenerate
progenitors may still play a role in producing short-delay SNe Ia, or perhaps
all SNe Ia, if the red-giant donor channel is more efficient than found by most
theoretical models. Apart from the DTD shape, the DTD normalization enjoys
fairly good agreement (though perhaps some tension), among the various
measurements, with a Hubble-time-integrated DTD value of about 2+/- 1 SNe Ia
per 1000 Msun (stellar mass formed with a low-mass-turnover IMF). A recent
attempt to characterize the local white dwarf binary population suggests that
the white dwarf merger rate can explain the Galactic SN Ia rate, if sub-Chandra
mergers lead to SN Ia events. We conclude by pointing to some future directions
that should lead to progress in the field, including measurement of the
bivariate (delay and stretch) SN Ia response function .
