  Parametrized motion planning algorithms have high degree of flexibility and
universality, they can work under a variety of external conditions, which are
viewed as parameters and form part of the input of the algorithm. In this paper
we analyse the parameterized motion planning problem in the case of sphere
bundles. Our main results provide upper and lower bounds for the parametrized
topological complexity; the upper bounds typically involve sectional categories
of the associated fibrations and the lower bounds are given in terms of
characteristic classes and their properties. We explicitly compute the
parametrized topological complexity in many examples and show that it may
assume arbitrarily large values.
