  We consider optimal transportation of measures on metric and topological
spaces in the case where the cost function and marginal distributions depend on
a parameter with values in a metric space. The Hausdorff distance between the
sets of probability measures with given marginals is estimated via the
distances between the marginals themselves. The continuity of the cost of
optimal transportation with respect to the parameter is proved in the case of
continuous dependence of the cost function and marginal distributions on this
parameter. Existence of approximate optimal plans continuous with respect to
the parameter is established. It is shown that the optimal plan is continuous
with respect to the parameter in case of uniqueness. Examples are constructed
when there is no continuous selection of optimal plans. Finally, a general
result on convergence of Monge optimal mappings is proved.
