This thesis contributes with a number of topics to the subject of string compactifications. In the first half of the work, I discuss the Hodge plot of Calabi-Yau threefolds realised as hypersurfaces in toric varieties. The intricate structure of this plot is explained by the existence of certain webs of elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along K3 slices. Any two half-polytopes over a given slice can be combined into a reflexiveâ€¦Â Expand