In 1981, Gromov asked whether there exist simply connected topological manifolds, other than Euclidean space, that admit a metric of non-positive curvature in a synthetic sense. Since CAT(0) spaces are contractible, it follows from the classification of surfaces that any CAT(0) 2-manifold is Euclidean. In dimension 3, by combining results of Brown and Rolfsen, CAT(0) manifolds are homeomorphic to R^3. Recently, Lytchak, Nagano, and Stadler proved that CAT(0) 4-manifolds are Euclidean. In this talk, I will discuss Gromov's question and introduce spaces of (global) non-positive curvature in the sense of Busemann, abbreviated as BNPC spaces. This notion is what Gromov originally intended by "synthetic sense." I will show that the results above can be extended to BNPC manifolds. This is joint work with Tadashi Fujioka.