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 dimcnsional effective theory, and in a quest to complete our understanding of the gcncrically torsional geometries that appear, it is important to understand the moduli of these geometries. I will bricfly review the infinitesimal moduli space in six dimensions, and comment on some rcccnt advances in understanding its geometrie properties. I will thcn cxplain how a similar story is emerging in seven dimensional compactifications on more exotic manifolds with G2-structure, giving hope that similar tools can be applicd 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.

DATA: 19-10-2016