  In this article, we summarize our recent results on the study of manifolds
with special holonomy via the Fr\"olicher-Nijenhuis bracket. This bracket
enables us to define the Fr\"olicher-Nijenhuis cohomologies which are analogues
of the $d^c$ and the Dolbeault cohomologies in K\"ahler geometry, and assigns
an $L_\infty$-algebra to each associative submanifold. We provide several
concrete computations of the Fr\"olicher-Nijenhuis cohomology.
