The space of holomorphic maps from
S
2
{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
)
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
)
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
)
→
∞
n(D) \to \infty
as
D
→
∞
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
)
n(D)
may be computed, and
n
(
D
)
→
∞
n(D) \to \infty
as
D
→
∞
D \to \infty
. For other singular toric varieties, however, it turns out that
n
(
D
)
n(D)
cannot always be made arbitrarily large by a suitable choice of D.