The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large N expansions. These topics are inter-related. For example, a third-order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second-order difference equation for the general complex moments $$M_{2p}^\textrm{r}$$
M
2
p
r
. The latter are expressed in terms of the $${}_3 F_2$$
3
F
2
hypergeometric functions, with a simplification to the $${}_2 F_1$$
2
F
1
hypergeometric function possible for $$p=0$$
p
=
0
and $$p=1$$
p
=
1
, allowing for the large N expansion of these moments to be obtained. The large N expansion involves both integer and half-integer powers of 1/N. The three-term recurrence then provides the large N expansion of the full sequence $$\{ M_{2p}^\textrm{r} \}_{p=0}^\infty $$
{
M
2
p
r
}
p
=
0
∞
. Fourth- and third-order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and the properties of the large N expansion of these quantities are determined.