We prove a cutoff for the random walk on random n-lifts of finite weighted graphs, even when the random walk on the base graph G of the lift is not reversible. The mixing time is w.h.p. t mix = h −1 log n, where h is a constant associated to G, namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all n-lifts of G. In the particular case where the base graph is a vertex with d/2 loops, d even, we obtain a cutoff for a d-regular random graph (as did Lubetzky and Sly in [26] with a slightly different distribution on d-regular graphs, but the mixing time is the same).