  We prove an extension of the Kato-Saito class field theory for smooth
projective schemes over a finite field to schemes with singularities. As an
application, we obtain Bloch's formula for the Chow groups of 0-cycles on such
schemes. We identify the Chow group of 0-cycles on a normal projective scheme
over an algebraically closed field to the Suslin homology of its regular locus.
Our final result is a Roitman torsion theorem for smooth quasi-projective
schemes over algebraically closed fields. This completes the missing
$p$-torsion part in the torsion theorem of Spiess and Szamuely.
