  $\mathcal{G}$-structures, where $\mathcal{G}$ is a Lie group, are a uniform
characterisation of many differential geometric structures of interest in
supersymmetric compactifications of string theories. Calabi-Yau $n$-folds are
instances of torsion-free $SU(n)$-structures, while more general structures
with non-zero torsion are required for heterotic flux compactifications.
Exceptional geometries in dimensions $7$ and $8$ with $\mathcal{G}=G_2$ and
$Spin(7)$ also feature prominently in this thesis. We discuss multiple
connections between such geometries and the worldsheet theory describing
strings on them, especially with respect to their chiral symmetry algebras.
