  In this paper we give a geometric characterization of the cones of toric
varieties that are complete intersections. In particular, we prove that the
class of complete intersection cones is the smallest class of cones which is
closed under direct sum and contains all simplex cones. Further, we show that
the number of the extreme rays of such a cone, which is less than or equal to
$2n-2$, is exactly $2n-2$ if and only if the cone is a bipyramidal cone, where
$n>1$ is the dimension of the cone. Finally, we characterize all toric
varieties whose associated cones are complete intersection cones.
