  In this thesis we study string compactifications on manifolds equipped with a
$G$-structure, placing a special emphasis on the interplay between geometry and
physics. We follow two complementary approaches. In the first part of the
thesis we adopt a sigma model perspective and focus on the worldsheet
superconformal field theory. We consider compactifications on 7-dimensional
Extra Twisted Connected Sum (ETCS) G$_2$ manifolds as well as 8-dimensional
Generalized Connected Sum (GCS) Spin(7) manifolds. We find that the geometric
construction is reproduced in the worldsheet algebra via a diamond of algebra
inclusions. In the second part of the thesis we change gears and consider
string compactifications from a supergravity point of view. In particular, we
focus on compactifications of the heterotic string down to three spacetime
dimensions preserving minimal supersymmetry $\mathcal{N}=1$, which are
described by the heterotic G$_2$ system. We construct new families of AdS$_3$
solutions to this system on homogeneous 3-Sasakian manifolds with squashed
metrics.
