: In the talk, we will study the Harnack inequality for weak solutions to elliptic/parabolic and kinetic equations with rough coefficients. This is often referred to as the De Giorgi-Nash-Moser theory. The proof of the Harnack inequality due to Moser (1971) is based on a weak L1-estimate for the logarithm of supersolutions combined with Lp-L?-estimates and a lemma due to Bombieri and Giusti. His method has been applied to nonlocal parabolic problems (Kassmann and Felsinger 2013), time-fractional diffusion equations (Zacher 2013), discrete problems (Delmotte 1999) and many more. In each of these works, the proof of the weak L1-estimate follows more or less the strategy of Moser and is based on a Poincaré inequality. I will present a novel proof of this weak L1-estimate for the logarithm of supersolutions which does not rely on any Poincaré inequality. The approach is based on trajectories induced by the geometry of the equation. This approach differs from Moser's strategy and gives a very nice geometric interpretation of the result. The argument does not treat the temporal and spatial variables separately but considers them simultaneously at the natural scale.