In this paper we study the determinant of elliptic differential operators on a complex vector bundle E→pM of rank N over a compact oriented connected manifold M of dimension 1, as well as the determinants of its finite difference approximations. For an elliptic differential operator A over S1, A=∑nk=0Ak(x)Dk, with Ak(x) in END(Cr) and θ as a principal angle, the ζ-regularized determinant DetθA is computed in terms of the monodromy map PA associated to A and some invariant expressed in terms of An and An−1. A similar formula holds for finite difference operators. A number of applications and implications are given. In particular, we present a formula for the signature of A when A is selfadjoint and show that the determinant of A is the limit of a sequence of computable expressions involving determinants of difference approximations of A.