I shall start by reporting what is known about $G_2$-instantons on noncompact $G_2$-manifolds. Then, I will focus on joint work with Jason Lotay concerning existence and classification results for these instantons. That work investigates the particular case of $R^4$ X $S^3$, with its two explicitly known distinct $G_2$-holonomy metrics, exhibiting the different existence/behaviour of $G_2$-instantons. We also give an explicit example of sequences of $G_2$-instantons where "bubbling" and "removable singularity" phenomena occur in the limit. If time permits, I will state some quite accessible (I hope) open problems. (This is joint work with Jason Lotay.)