  $G_2$-Monopoles are solutions to gauge theoretical equations on noncompact
$7$-manifolds of $G_2$ holonomy. We shall study this equation on the $3$
Bryant-Salamon manifolds. We construct examples of $G_2$-monopoles on two of
these manifolds, namely the total space of the bundle of anti-self-dual two
forms over the $\mathbb{S}^4$ and $\mathbb{CP}^2$. These are the first
nontrivial examples of $G_2$-monopoles. Associated with each monopole there is
a parameter $m \in \mathbb{R}^+$, known as the mass of the monopole. We prove
that under a symmetry assumption, for each given $m \in \mathbb{R}^+$ there is
a unique monopole with mass $m$. We also find explicit irreducible
$G_2$-instantons on $\Lambda^2_-(\mathbb{S}^4)$ and on
$\Lambda^2_-(\mathbb{CP}^2)$. The third Bryant-Salamon $G_2$-metric lives on
the spinor bundle over the $3$-sphere. In this case we produce a vanishing
theorem for monopoles.
