Finite edge-transitive bi-circulants

•We have considered carefully each comment of the reviewer, and made corresponding changes in the revision.•We believe these changes are correct and suitable.•We have also improved some presentations of the paper. This work studies edge-transitive bi-circulants of order 2n with Zn being a Hall subgroup of the full automorphism groups. The automorphism groups of such graphs are completely determined. As an application, a classification is given of bipartite edge-transitive bi-circulants of order 2n with valency less than the smallest prime divisors of n. This also leads to a new construction of an infinite family of arc-regular bi-circulants.