  The paper is concerned with the problem on rolling of a homogeneous ball on
an arbitrary surface. New cases when the problem is solved by quadratures are
presented. The paper also indicates a special case when an additional integral
and invariant measure exist. Using this case, we obtain a nonholonomic
generalization of the Jacobi problem for the inertial motion of a point on an
ellipsoid. For a ball rolling, it is also shown that on an arbitrary cylinder
in the gravity field the ball's motion is bounded and, on the average, it does
not move downwards. All the results of the paper considerably expand the
results obtained by E. Routh in XIX century.
