The transportation problem is the following. Given a permutation action of a group G on a set S and two elements of S, find one or all of the group elements transporting one element to the other. Perhaps this is the most natural common generalization of computing discrete logarithms and graph isomorphisms. The hidden shift problem for injective functions is essentially equivalent to the special case when the action is regular. We discuss the best known quantum algorithms for the tasks above and related problems such as finding hidden subgroups and variants of the shift problem in certain groups. Some approaches rely on solving relaxations of simple but hard arithmetic problems. Progress in solving these subtasks could result in faster quantum algorithms for some of the transportation-related problems.