The construction of manifolds of non-negative (or positive) sectional curvature is intimately tied to Lie group actions on manifolds. Some classical examples are symmetric spaces, space forms and homogeneous spaces. In this talk we will see the history of constructing examples using group actions and their importance for non-negative curvature. While there are obstructions like Gromov's Betti number theorem, the gap between obstructions and known examples remains quite wide. We will look at some recent ideas to produce examples of new topological types. We will also mention some open problems along the way.