  Chaplygin proved the integrability by quadratures of a round sphere, rolling
without slipping on a horizontal plane, with center of mass at the center of
the sphere, but with arbitrary moments of inertia. Although the system is
integrable in every sense of the word, it neither is a hamiltonian system, nor
is the integrability an immediate consequence of the symmetries. On the other
hand the constants of motion are obtained as a consequence of a Nother's
principle and the system can be related to the geodesic flow on the Euclidean
motion group for a left invariant metric.
  In this paper we analyse the global dynamics and in the process we will
explain almost all of Chaplygin's results. At the end of each section we
describe in a subsection ``Chaplygin'' the relation between our text and
Chaplygin's.
  We also obtain several new results, such as the proof that the level sets of
the constants of motion in the phase space for the rotational motion are
two-dimensional tori on which, after a suitable time reparametrization, the
rotational motion is quasiperiodic. After suitable completion of the level
surfaces this is also true for the complexified system, which is algebraically
integrable in the sense of Adler and van Moerbeke. This also follows, in a
quite different way, from Chaplygin's integration in terms of hyperelliptic
integrals.
