  I discuss the use of Type Ia supernovae (SNe Ia) for cosmological distance
determinations. Low-redshift SNe Ia (z < 0.1) demonstrate that the Hubble
expansion is linear with H_0 = 72 +/- 8 km/s/Mpc, and that the properties of
dust in other galaxies are generally similar to those of dust in the Milky Way.
The measured luminosity distances of SNe Ia as a function of redshift have
shown that the expansion of the Universe is currently accelerating, probably
due to the presence of repulsive dark energy such as Einstein's cosmological
constant (Lambda). From about 200 SNe Ia, we find that Omega_Lambda - 1.4
Omega_M = 0.35 +/- 0.14. Combining our data with other results, we find a best
fit for Omega_M and Omega_Lambda of 0.28 and 0.72, respectively. A number of
possible systematic effects (dust, supernova evolution) thus far do not seem to
eliminate the need for Omega_Lambda > 0. Recently, analyses of SNe Ia at z =
1.0-1.7 provide further support for current acceleration, and give tentative
evidence for an early epoch of deceleration. The dynamical age of the Universe
is estimated to be 13.1 +/- 1.5 Gyr. According to the most recent data sets,
the SN Ia rate at z > 1 is several times greater than that at low redshifts,
presumably because of higher star formation rates long ago. Moreover, the
typical delay time from progenitor star formation to SNIa explosion appears to
be substantial, ~3 Gyr. Current projects include the measurement of a few
hundred SNe Ia at z = 0.2-0.8 to more accurately determine the
equation-of-state parameter of the dark energy, w = P/(\rho c^2), whose value
is now constrained by SNe Ia to be in the range -1.48 < w < -0.72 at 95%
confidence.

Astrophysics