An extensive analysis has been done on the numerical range of an operator, however, little research has been done on its real analogue. In this thesis we give a number of results and properties regarding the real numerical range, and real higher rank numerical range. We motivate this study by providing the reader with an application of how the real higher rank numerical range may be used in the study of conic sections. Finally, we end the paper with a short introduction into the eld of quantum information theory, eventually building up to introduce a new measure of entanglement for pure symmetric states.