  We review the use of Type Ia supernovae 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. We find that 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 luminosity distances of our first set of 16 high-redshift
SNe Ia are, on average, 10-15% farther than expected in a low mass-density
(Omega_M=0.2) universe without a cosmological constant. Preliminary analysis of
our second set of 9 SNe Ia is consistent with this. Our work supports models
with positive cosmological constant and a current acceleration of the
expansion. We address the main potential sources of systematic error; at
present, none of them appears to reconcile the data with Omega_Lambda=0 and q_0
>= 0. The dynamical age of the Universe is estimated to be 14.2 +/- 1.7 Gyr,
consistent with the ages of globular star clusters.
