  We construct self-intersected flexible cross-polytopes in the spaces of
constant curvature, that is, the Euclidean spaces, the spheres, and the
Lobachevsky spaces of all dimensions. In dimensions greater than or equal to 5,
these are the first examples of flexible polyhedra. Moreover, we classify all
flexible cross-polytopes in each of the spaces of constant curvature. For each
type of flexible cross-polytopes, we provide an explicit parametrization of the
flexion by either rational or elliptic functions.
