We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank
2
2
” presentation for the group of
F
F
-rational points of an arbitrary exceptional simple group of
F
F
-rank at least
4
4
and to determine defining relations for the group of
F
F
-rational points of an an arbitrary group of
F
F
-rank
1
1
and absolute type
D
4
D_4
,
E
6
E_6
,
E
7
E_7
or
E
8
E_8
associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.