The Geometry and Superconformal Algebras of String Compactifications with a $G$-structure

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.