  The paper concerns a simple model of bicycle kinematics: a bicycle is
represented by an oriented segment of constant length in n-dimensional space
that can move in such a way that the velocity of its rear end is aligned with
the segment (the rear wheel is fixed on the bicycle frame). Starting with a
closed trajectory of the rear end, one obtains the two respective trajectories
of the front end, corresponding to the opposite directions of motion. These two
curves are said to be in the bicycle correspondence. Conjecturally, this
correspondence is completely integrable; we present a number of results
substantiating this conjecture. In dimension three, the bicycle correspondence
is the Backlund transformation for the filament equation; we discuss
bi-Hamiltonian features of the bicycle correspondence and its integrals.
