Calabi-Yau Hypersurfaces and Complete Intersections in Toric Varieties

In this thesis we report on several projects that stemmed out from an attempt to obtain an example for the last class in Doran-Morgan classification of variations of Hodge structure which can underlie families of Calabi-Yau threefolds over the thrice-punctured sphere with b3 = 4, or equivalently h2,1 = 1 [DM06]