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, thatH0=65±2(statistical) km−1Mpc−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 of1+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&percent; farther than expected in a low mass-density(&OHgr;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&OHgr;&Lgr;=0andq0⩾0.The dynamical age of the Universe is estimated to be14.2±1.7&hthinsp;Gyr,consistent with the ages of globular star clusters. ©2000 American Institute of Physics.