Abstract:  I will discuss recent advances in addressing the heterotic moduli problem in six and seven dimensional compactifications.  To map out the reduction to a lower dimensional effective theory, and in a quest to complete our understanding of the generically torsional geometries that appear, it is important to understand the moduli of these geometries.  I will briefly review the infinitesimal moduli space in six dimensions, and comment on some recent advances in understanding its geometric properties.  I will then explain how a similar story is emerging in seven dimensional compactifications on more exotic manifolds with G2-structure, giving hope that similar tools can be applied in other areas of string theory to understand moduli.  If time, I will comment on some recent advances is studying finite deformations and obstructions of the moduli space.