This thesis develops the deformation theory of instantons on asymptotically
conical G2-manifolds, where an asymptotic connection at infinity is fixed. A
spinorial approach is adopted to relate the space of deformations to the kernel
of a twisted Dirac operator on the G2-manifold and to the eigenvalues of a
twisted Dirac operator on the nearly Kähler link. As an application, we use
this framework to study the moduli spaces of known examples of G2-instantons
living on the Bryant-Salamon manifolds and on R7. We develop two methods
for determining eigenvalues of twisted Dirac operators on nearly Kähler 6-
manifolds and apply this to calculate the virtual dimension of the moduli
spaces that we study. In the case of the instanton of Günaydin-Nicolai, which
lives on R7; we show how knowledge of the virtual dimension of the moduli
space can be used to study uniqueness properties of this instanton
