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When string theory is compactified on a manifold M, we can approach the resulting theory from three perspectives: the geometry of M, the worldsheet theory of strings probing M (typically a superconformal field theory), or the effective theory in the remaining spacetime directions. In the case that M was a Calabi-Yau manifold, it turned out to be very productive to study the connections between these points of view, leading for instance to the discovery of mirror symmetry. When M is a manifold of G_2 or Spin(7) holonomy, the problem has been much less well-studied. We will describe improvement in this situation and prospects for the future.