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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[sub 0] = 65 ± 2 (statistical) km s[sup -1] Mpc[sup -1], 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 massdensity (Ω[sub 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 Ω[sub Lambda] = 0 and q[sub 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. [ABSTRACT FROM AUTHOR]