The theory of vegetation patterns presented rests on two hypotheses: (i) the self-organization hypothesis that attributes their cause to interactions intrinsic to vegetation dynamics; (ii) the complementary self-assembly hypothesis that attributes their large spatial scale to the proximity of their dynamical conditions with a critical point. A non-local version of the F–KPP equation allows us to formulate these hypotheses in terms of individual plant properties. Both general and parsimonious, this formulation is strictly quantitative. It only relies on structural parameters that can be measured with precision in the field. Quantitative interpretation of observations and of the predictions provided by the theory is illustrated by an analysis of the periodic patterns found in some Sub-Sahelian regions.