  The space of holomorphic maps from $S^2$ to a complex algebraic variety $X$,
i.e. the space of parametrized rational curves on $X$, arises in several areas
of geometry. It is a well known problem to determine an integer $n(D)$ such
that the inclusion of this space in the corresponding space of continuous maps
induces isomorphisms of homotopy groups up to dimension $n(D)$, where $D$
denotes the homotopy class of the maps. The solution to this problem is known
for an important but special class of varieties, the generalized flag
manifolds: such an integer may be computed, and $n(D)\to\infty$ as
$D\to\infty$. We consider the problem for another class of varieties, namely,
toric varieties. For smooth toric varieties and certain singular ones, $n(D)$
may be computed, and $n(D)\to\infty$ as $D\to\infty$. For other singular toric
varieties, however, it turns out that $n(D)$ cannot always be made arbitrarily
large by a suitable choice of $D$.
