Torsion of Elliptic Curves

This dissertation describes two papers, one published and one in preparation, about the torsion subgroups of Elliptic Curves. The first chapter gives a brief introduction to elliptic curves and the history of research on torsion points of elliptic curves. The second chapter establishes a concept of typical boundedness for the torsion groups of a family of elliptic curves. The third chapter gives a partial classification for torsion groups of elliptic curves with complex multiplication defined over Abelian number fields.