  In this work we provide a self-contained and modern introduction to some of
the tools, obstacles and open questions arising in string compactifications.
Techniques and current progress are illustrated in the context of smooth
heterotic string compactifications to 4-dimensions. Progress is described on
bounding and enumerating possible string backgrounds and their properties. We
provide an overview of constructions, partial classifications, and moduli
problems associated to Calabi-Yau manifolds and holomorphic bundles over them.
