Manifolds with $G$-structures are suitable as internal manifolds for supersymmetric flux compactifications, once the backreaction of fluxes on the internal geometry is taken into account. After a general introduction to flux compactifications, I will focus on supersymmetric type IIA compactifications on $SU(3)$ structure manifolds, where necessary and sufficient conditions on the internal geometry have been found. As an illustration, I will discuss the Nilsson-Pope example, where the internal manifold is a $\mathbb{CP}3$. Noting that $\mathbb{CP}3$ is a toric variety, I will outline a generalization to other toric varieties, which could also be interesting for string compactifications.