  A complete bipartite graph $K_{3,3}$, considered as a planar linkage with
joints at the vertices and with rods as edges, in general admits only motions
as a whole, i.e., is inflexible. Two types of its paradoxical mobility were
found by Dixon in 1899. Later on, in a series of papers by different authors,
the question of flexibility of $K_{m,n}$ was solved for almost all pairs
$(m,n)$. In the present paper, we solve it for all complete bipartite graphs in
the Euclidean plane as well as in the sphere and in the hyperbolic plane. We
give independent self-contained proofs without extensive computations which are
almost the same in the Euclidean, hyperbolic and spherical cases.
