We present a method for tabulating all cubic function fields over
F
q
(
t
)
\mathbb {F}_q(t)
whose discriminant
D
D
has either odd degree or even degree and the leading coefficient of
−
3
D
-3D
is a non-square in
F
q
∗
\mathbb {F}_{q}^*
, up to a given bound
B
B
on
deg
(
D
)
\deg (D)
. Our method is based on a generalization of Belabas’ method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires
O
(
B
4
q
B
)
O(B^4 q^B)
field operations as
B
→
∞
B \rightarrow \infty
. The algorithm, examples and numerical data for
q
=
5
,
7
,
11
,
13
q=5,7,11,13
are included.