Einstein's theory of gravity (general relativity, GR) is based on Riemannian geometry. Modified theories of gravity such as f(R) theories are also based on Riemann spaces and so are intrinsically limited. In order to better understand the nature of gravity it is worth going further, beyond Riemannian geometry. The simplest extensions of Riemannian geometry are based on non-null non-metricity tensors and non-symmetric connections. I will briefly discuss these extensions and focus on models with a totally antisymmetric torsion tensor, i.e. with a null non-metricity tensor. I will show how the presence of torsion affects light propagation and I will discuss how to use cosmological observations to test some of these non-Riemannian extensions of the geometry of our Universe.