It is a classical question to examine which smooth manifolds admit Riemannian metrics with positive/non-negative sectional curvature. In this talk, I will discuss some joint work with Manuel Amann on this question in the presence of torus symmetry, specifically in small dimensions. It extends work of Dessai, where a number of topological invariants of 8-manifolds with positive curvature and torus symmetry are calculated, both in general and under additional assumptions (e.g., rationally elliptic or homogeneous).