I will give a (biased) overview of recent progress on the construction of complete noncompact metrics of exceptional holonomy. Along the way I will describe some of the most important historical developments since the field began (in the late 1970s). Throughout its history, the field has seen a fruitful back-and-forth between physicists and mathematicians, some of which I will describe. I will try to explain some of the similarities and differences between the more familiar special holonomy metrics — hyperkaehler and Calabi—Yau metrics— and the exceptional cases  and  holonomy, and why the latter are much more difficult to construct.

In the early 2000s M theorists predicted the existence of various new complete noncompact Riemannian metrics with holonomy group the compact exceptional Lie group G2. Recently mathematicians have constructed many, but by no means all, of these physically predicted metrics and also other  metrics not necessarily anticipated by physics. It will turn out the construction of these complete noncompact metrics of exceptional holonomy relies on some of the most recent developments on constructing complete noncompact hyperkaehler and Calabi—Yau metrics with controlled asymptotic geometry