The classification of almost non-negatively curved, simply connected, closed manifolds is an open and difficult problem. The Symmetry Program suggests a means to approach such a classification. In this thesis we consider almost non-negatively curved n-manifolds, 4 ≤ n ≤ 6, with torus symmetry. We obtain a classification up to equivariant diffeomorphism for 4 ≤ n ≤ 6 when the action is of maximal symmetry rank. We also obtain a partial classification of the almost maximal symmetry rank case for n = 5, noting that a complete classification for n = 4 was obtained by Harvey and Searle.