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This thesis is comprised of three chapters. The first chapter deals withbounded complexes of Gorenstein projective and Gorenstein injective modules.Deploying methods of relative homological algebra, we approximate suchcomplexes with bounded complexes of projective and injective modules,respectively. As an application, we investigate the Gorenstein version of theNew Intersection Theorem.  The second chapter studies the notion of cofiniteness for modules andcomplexes set forth by Hartshorne. Recruiting techniques of derived category,we study this notion thoroughly, obtain novel results, and extend some of theresults to stable under specialization sets.  The third chapter delves into the Greenlees-May Duality Theorem which iswidely thought of as a far-reaching generalization of the Grothendieck's LocalDuality Theorem. This theorem is not addressed in the literature as it meritsand its proof is indeed a tangled web in a series of scattered papers. Bycarefully scrutinizing the requisite tools, we present a clear-cutwell-documented proof of this theorem.

Gorenstein Homology and Finiteness Properties of Local (Co)homology