The reader of this thesis should already have a basic understanding of ideal theory. For this reason it is recommended that a good introduction to this subject would be gained from reading D. G. Northcott's book "Ideal Theory", paying special attention to chapters one and three. This thesis consists of three chapters, with chapter one providing the definitions and theorems which will be used throughout. Then I will be considering two problems on the arithmetic degree of an ideal, one posed by Sturmfels, Trung and Vogel and the other by Renschuch. These problems will be described in the introductions to chapters two and three.