We study complete varieties on which a connected solvable group acts with
finitely many orbits, for example toric varieties or Schubert varieties.
For these varieties, we exhibit a finite presentation of their Chow groups in
terms of the orbit closures and compute the operational Chow cohomology
ring. For smooth varieties of this type, their cohomology ring and rich
combinatorial structure have been vital in many areas of mathematics. Our
primary goal is to extend this as far as is possible to the singular case.