This thesis contains two main areas of research in General Relativity Theory. These are the study of the sectional curvature function in general relativity and the study of symmetries. The sectional curvature function is a real-valued map defined on the set of all non-null 2-spaces at a certain point in the space-time. several results relating to the sectional curvature function will be given. The bivector curvature function will then be defined as the extension of the sectional curvature function to the set of all "non-null" bivectors at a point in the space-time. Two important results relating to this function will be proved. Symmetries in general relativity have been widely researched. In this thesis, three results on symmetries will be proved. Firstly, it will be shown that there exists a space-time admitting a finite-dimensional curvature collineation algebra not equal to the affine algebra. Then a result on the conformal algebra in a 2-dimensional manifold will be given. Lastly, a proof will be given on the dimension of the sectional curvature preserving algebra.