  I review 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, that H_0 = 65 +/- 2 (statistical) km/s/Mpc, and that the
properties of dust in other galaxies are similar to those of dust in the Milky
Way. The light curves of high-redshift (z = 0.3 - 1) SNe Ia are stretched in a
manner consistent with the expansion of space; similarly, their spectra exhibit
slower temporal evolution (by a factor of 1 + z) than those of nearby SNe Ia.
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). Combining our data with existing measurements of the cosmic
microwave background (CMB) radiation and with the results of large-scale
structure surveys, we find a best fit for \Omega_m and \Omega_\Lambda of about
0.3 and 0.7, respectively. Other studies (e.g., masses of clusters of galaxies)
also suggest that \Omega_m \approx 0.3. The sum of the densities, ~1.0, agrees
with the value predicted by most inflationary models for the early Universe:
the Universe is flat on large scales. A number of possible systematic effects
(dust, supernova evolution) thus far do not seem to eliminate the need for
\Omega_\Lambda > 0. Most 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. Current projects include the search for additional
SNe Ia at z > 1 to confirm the early deceleration, and the measurement of a few
hundred SNeIa at z = 0.2 - 0.8 to determine the equation of state of the dark
energy, w = P/(\rho c^2).
