  This thesis aims to give the reader an introduction and overview of the
cd-index of a poset, as well as establish some new results. We give a
combinatorial proof of Ehrenborg and Karu's cd-index subdivision decomposition
for Gorenstein* complexes and extend it to a wider class of subdivisions. In
doing so, we define a local cd-index that behaves analogously to the well
studied local h-vector. We examine known cd-index and h-vector bounds, and then
use the local cd-index to bound a particular class of polytopes with the
cd-index of a stacked polytope. We conclude by investigating the h-vector and
local h- vector of posets in full generality, and use an algebra morphism
developed by Bayer and Ehrenborg to demonstrate the structural connection
between the cd-index subdivision decomposition and the local h-vector
subdivision decomposition. iii
