In this paper we study θ-cyclic codes over the ring R = F 2 + vF 2 = {0, 1, v, v +1} where v 2 = v. This is the only ring of order four that is not a field and has a non-trivial ring automorphism. We describe generator polynomials of θ-cyclic codes defined over this ring. We also describe the generator polynomials of the duals of free θ-cyclic codes with respect to Euclidean and Hermitian inner products. Finally, we give examples of optimal self-dual codes with respect to the Euclidean and Hermitian inner products.