We prove that the rational homology Betti numbers of a toric variety with singularities are not necessarily determined by the combinatorial type of the fan which defines it; that is, the homology is not determined by the partially ordered set formed by the cones in the fan. We apply this result to the study of convex polytopes, giving examples of two combinatorially equivalent polytopes for which the associated toric varieties have different Betti numbers. Our main result is that the rational homology Betti numbers of a toric variety with singularities are not necessarily determined by the combinatorial type of the fan which defines it; that is, the homology is not determined by the partially ordered set formed by the cones in the fan. This holds in all dimensions n > 3. The result is in contrast with the following facts: (1) The rational homology Betti numbers of a nonsingular toric variety are determined by the combinatorial type of the fan. The rational cohomology ring of nonsingular toric varieties played a central role in the proof of McMullen's conjecture concerning the number of faces of simplicial convex polytopes [S1, BL]. (2) The intersection homology Betti numbers of a singular toric variety are determined by the combinatorial type of the fan. This fact provides information about general rational convex polytopes [S2]. We give an algorithm (1.2-1.3) for computing the Betti numbers of a complete toric variety of dimension three. We then give examples (1.4) of two combinatorially equivalent polyhedra for which the associated toric varieties have different Betti numbers. In ?2 we prove the results in (1.2-1.3), and in ?3 we conclude with a few remarks. 1. STATEMENT OF RESULTS (1.1) Let a denote any closed convex rational polyhedral cone in Rn which does not contain a line. A (complete) fan I is a finite collection {la} which Received by the editors April 22, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 52A25, 14L32.