A free n-Engel group is the relatively free group of the variety of groups with the identical relation [x, y, y,...,y (n times)]=1. Let n>=20. We show that the free Engel group on at least two generators is not locally nilpotent. Our approach to Engel groups combines
geometric and combinatorial methods. On the geometric side, we consider graded van Kampen diagrams, and we prove that they display (discrete) negative curvature properties. To do this, we construct a canonical form of elements in each consecutive rank (this is the combinatorial aspect). Using the canonical form, we obtain "parallel meetings" between the regions of higher ranks of the graded van Kampen diagram and, using surgery, improve it to direct meeting. The combinatorial structure of the relators secures that this direct meeting is (relatively) short. Given the structure of graded van Kampen diagrams, we deduce a graded version of Greendlinger's Lemma and then establish the properties of the group.