We undertake a study of bijections which are used to enumerate sets of permutations and labeled forests according to various statistics. A permutation statistic is called Mahonian if it has the same distribution on the symmetric group $S\sb{n}$ as the inversion statistic. The major index and inversion index are the fundamental examples of Mahonian statistics. The inversion index has been extended by Mallows and Riordan to labeled forests. Recently, Bjorner and Wachs generalized the major index to labeled forests and showed that the major index has the same distribution as the inversion index on labeled forests of fixed shape. We give a direct combinatorial proof of this result by constructing an explicit bijection on labeled forests which takes the major index to the inversion index. For the symmetric group this bijection reduces to a new bijection on $S\sb{n}$ taking the major index to the inversion index which is similar to a bijection of Foata. We also generalize the Mahonian statistics of Rawlings and Kadell to labeled forests and show that they have the same distribution as the inversion index as well. Generalizations of the Foata bijection on permutations are also presented.