  We study a simple model of bicycle motion: a segment of fixed length in
multi-dimensional Euclidean space, moving so that the velocity of the rear end
is always aligned with the segment. If the front track is prescribed, the
trajectory of the rear wheel is uniquely determined via a certain first order
differential equation -- the bicycle equation. The same model, in dimension
two, describes another mechanical device, the hatchet planimeter.
  Here is a sampler of our results. We express the linearized flow of the
bicycle equation in terms of the geometry of the rear track; in dimension
three, for closed front and rear tracks, this is a version of the Berry phase
formula. We show that in all dimensions a sufficiently long bicycle also serves
as a planimeter: it measures, approximately, the area bivector defined by the
closed front track. We prove that the bicycle equation also describes rolling,
without slipping and twisting, of hyperbolic space along Euclidean space. We
relate the bicycle problem with two completely integrable systems: the AKNS
(Ablowitz, Kaup, Newell and Segur) system and the vortex filament equation. We
show that "bicycle correspondence" of space curves (front tracks sharing a
common back track) is a special case of a Darboux transformation associated
with the AKNS system. We show that the filament hierarchy, encoded as a single
generating equation, describes a 3-dimensional bike of imaginary length. We
show that a series of examples of "ambiguous" closed bicycle curves (front
tracks admitting self bicycle correspondence), found recently F. Wegner, are
buckled rings, or solitons of the planar filament equation. As a case study, we
give a detailed analysis of such curves, arising from bicycle correspondence
with multiply traversed circles.
