  Dropping separatedness in the definition of a toric variety, one obtains the
more general notion of a toric prevariety. Toric prevarieties occur as ambient
spaces in algebraic geometry and moreover they appear naturally as intermediate
steps in quotient constructions. We first provide a complete description of the
category of toric prevarieties in terms of convex-geometrical data, so-called
systems of fans. In a second part, we consider actions of subtori H of the big
torus of a toric prevariety X and investigate quotients for such actions. Using
our language of systems of fans, we characterize existence of good prequotients
for the action of H on X. Moreover, we show by means of an algorithmic
construction that there always exists a toric prequotient for the action of H
on X, that means an H-invariant toric morphism p from X to a toric prevariety Y
such that every H-invariant toric morphism from X to a toric prevariety factors
through p. Finally, generalizing a result of D. Cox, we prove that every toric
prevariety occurs as the image of a categorical prequotient of an open toric
subvariety of some complex affine space.
