  We review the use of Type Ia supernovae for cosmological distance
determinations. Low-redshift SNe Ia (z < 0.1) demonstrate that (a) the Hubble
expansion is linear, (b) H_0 = 65 +/- 2 (statistical) km/s/Mpc, (c) the bulk
motion of the Local Group is consistent with the COBE result, and (d) 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 strongly supports
eternally expanding models with positive cosmological constant and a current
acceleration of the expansion. We address many potential sources of systematic
error; at present, none of them reconciles 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.
