In this thesis, we study combinatorial and D-module theoretic aspects of local cohomology. Viewing local cohomology from the point of view of A-hypergeometric systems, the quasidegree set of the non-top local cohomology modules supported at the maximal ideal m of a toric ideal determine parameters ? where the rank of the corresponding hypergeometric system is higher than expected. We discuss the Macaulay2 package, Quasidegrees, and its main functions. The main purpose of Quasidegrees is to compute where the rank of the solution space of an A-hypergeometric system is higher than expected. Local duality gives a vector space isomorphism between local cohomology and Ext-modules. However, the proof of local duality is nonconstructive. We recall a combinatorial construction by Irena Peeva and Bernd Sturmfels to minimally resolve codimension 2 lattice ideals. With motivation coming from A-hypergeometric systems, we use the construction by Peeva and Sturmfels to construct an explicit local duality isomorphism for codimension 2 lattice ideals. In general, local cohomology modules of a ring may not be finitely generated. However, they still may possess other finiteness properties. In 1990, Craig Huneke asked if the number of associated prime ideals of a local cohomology module is finite. Using characteristic free D-module techniques inspired by Glennady Lyubeznik, we answer Huneke's question in the affirmative for local cohomology modules over Stanley-Reisner rings.
