Quintic threefolds are some of the simplest examples of Calabi-Yau varieties. An interesting relationship, discovered by string theorists, is that every Calabi-Yau variety Y has a mirror Calabi-Yau variety Y. In fact mirror symmetry is a relationship which relates complex structure moduli space of Y to the complexified Kahler moduli of its mirror Y. The purpose of this dissertation is to describe the complex structure moduli space from the point of view of geometric invariant theory (GIT).;The GIT compactification of quintic threefolds consists of adding certain singular quintic threefolds to the space of smooth quintic threefolds. An explicit description of the allowed singularities for this moduli space will be described. The description of allowed singularities is arrived at by a combinatorial procedure described by Mukai [15]. His method can be used to find the maximal semistable families of the moduli space. These maximal semistable families give a description of the possible singularities which can occur in the moduli space. The boundary structure of the compactification is also described in this dissertation.
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