Madero-Craven, Monica GraceA numerical semigroup is a subset, S of the non-negative integers, Z+ which contains zero, is closed under addition, and whose complement in Z+ is finite. We discuss the basic properties of numerical semigroups as well as associated structures such as relative ideals. Further, we examine several finite subsets of S including the Apery Set and two of its subsets. Relationships between these subsets of S will allow us to give an equivalent definition for S to be symmetric as well as a necessary condition for S to be almost symmetric.en-USThesisMathematics