It has recently been realised that polystable, holomorphic sums of line bundles over smooth Calabi-Yau three-folds provide a fertile ground for heterotic model building. Large numbers of phenomenologically promising such models have been constructed for various classes of Calabi-Yau manifolds. I will briefly discuss the class of models based on complete intersections in products of projective spaces. I will also present a particular line bundle model constructed on the tetra-quadric manifold. Further, I will explore the embedding of the line bundle sum into the larger moduli space of non-Abelian bundles, both by means of constructing specific polystable non-Abelian bundles and by turning on VEVs in the associated low-energy theory. The non-Abelian compactifications thus constructed lead to SU(5) GUT models with an extra global U(1) symmetry, which combined with the hypercharge leads to a B-L symmetry. The non-Abelian compactifications inherit many of the appealing phenomenological features of the Abelian model, such as the absence of dimension four and dimension five operators triggering a fast proton decay.