In their seminal paper [DT98] Donaldson–Thomas pointed out the possibility of an enumerative invariant for G2–manifolds obtained by counting certain connections, called G2–
instantons. This putative invariant is sometimes referred to as the G2 Casson invariant,
since it should be formally similar to the Casson invariant for 3–manifolds. In this thesis I
prove existence results for G2–instantons on G2–manifolds arising from Joyce’s generalised
Kummer construction [Joy96b, Joy00] as well as the twisted connected sum construction
[Kov03,CHNP12b]. These yield a number of concrete examples of G2–instantons and may,
in the future, help to compute the G2 Casson invariant. Moreover, I show how to construct
families of G2–instantons that bubble along associative submanifolds. From this construction
it follows that a naïve count of G2–instantons cannot yield a deformation invariant of
G2–manifolds. Nevertheless, there can still be hope for a G2 Casson invariant by counting
G2–instantons as well as associative submanifolds (and objects in between) with carefully
chosen weights. I present a promising proposal for the definition of these weights in the low
energy SU(2)–theory.