Abstract: I will address the problem of classifying supersymmetric string compactifications in regions of moduli space where the supergravity approximation breaks down. I will show evidence that, in the heterotic string, quantum effects generically turns a Calabi-Yau manifold into a T-fold, a non-geometric space defined by T-duality transition functions. I will describe a non-geometric version of ADE singularities, and show a full classification of the six-dimensional super-conformal field theories that describe the low energy physics on such defects.