  We compare various groups of 0-cycles on quasi-projective varieties over a
field. As applications, we show that for certain singular projective varieties,
the Levine-Weibel Chow group of 0-cycles coincides with the corresponding
Friedlander-Voevodsky motivic cohomology. We also show that over an
algebraically closed field of positive characteristic, the Chow group of
0-cycles with modulus on a smooth projective variety with respect to a reduced
divisor coincides with the Suslin homology of the complement of the divisor. We
prove several generalizations of the finiteness theorem of Saito and Sato for
the Chow group of 0-cycles over $p$-adic fields. We also use these results to
deduce a torsion theorem for Suslin homology which extends a result of Bloch to
open varieties.
