In this paper, Mahler gives two interesting results related to Liouville's rational approximation theorem for algebraic numbers. First, he proves the analogue of Liouville's result for function fields with coefficients in an arbitrary field. Second, he shows that, in contrast to the situation for fields of characteristic zero, Liouville's theorem for algebraic functions cannot be improved if the ground field is of finite characteristic. \par Reprint of the author's paper [Can. J. Math. 1, 397--400 (1949; Zbl 0033.35203)].