During the last decade, different theories have been proposed for developing a first order analysis on metric measure spaces. The common idea underpinning some of these non-linear theories is that, for a viable theory of first order calculus in this abstract setting, one needs plenty of curves well distributed along the space. One way of making this idea precise is to assume that the space supports a p-Poincaré inequality. In this talk, we will review some of the latest results which have contributed to understand the geometrical structure of metric measure spaces supporting a p-Poincaré inequality and discuss purely geometric characterizations of p-Poincaré inequality for different values of the exponent p.