Carleson has proved that the Fourier series of functions belonging to the class L² converge almost everywhere.Improving his method, Hunt proved that the Fourier series of functions belonging to the class L^p (p > 1) converge almost everywhere. On the other hand, Kolmogoroff proved that there is an integrable function whose Fourier series diverges almost everywhere. We shall generalise Kolmogoroff's Theorem as follows: There is a function belonging to the class L(logL)^p (p > 0) whose Fourier series diverges almost everywhere. The following problem is still open: whether “almost everywhere” in the last theorem can be replaced by “everywhere” or not. This problem is affirmatively answered for the class L by Kolmogoroff and for the class L(log logL)^p (0 < p < 1) by Tandori.