Geometry and incompleteness of G_2-moduli spaces

Riemannian manifolds with holonomy G_2 form an exceptional class of Ricci-flat manifolds occurring only in dimension 7. In the compact setting, their moduli spaces are known to be smooth (unobstructed), finite-dimensional, and to carry a natural Riemannian structure induced by the L^2-metric; but besides this very little is known about the global properties of G_2-moduli spaces. In this talk, I will review the basics of G_2-geometry and present new results concerning the distance theory and the geometry of the moduli spaces.