One of the main contributors to the vastness of the string landscape is the immense number of Calabi-Yau (CY) manifolds on which the theory can be compactified. Currently, one of the largest sets of CYs is obtained from hypersurfaces in toric varieties, which result from fine, regular, star triangulations (FRSTs) of reflexive polytopes. In this talk I will present new developments in the study of FRSTs. I will describe how the space of FRSTs is connected and a derivation for a new upper bound for the total number of FRSTs, and hence for the number of hypersurface CYs. I will also discuss prospects for determining what a "typical" triangulation (and CY) looks like and the counting-measure predictions that could be made from that.