  In this paper the authors show how to use Riemann-Hilbert techniques to prove
various results, some old, some new, in the theory of Toeplitz operators and
orthogonal polynomials on the unit circle (OPUC's). There are four main
results: the first concerns the approximation of the inverse of a Toeplitz
operator by the inverses of its finite truncations. The second concerns a new
proof of the `hard' part of Baxter's theorem, and the third concerns the Born
approximation for a scattering problem on the lattice $\mathbb{Z}_+$. The
fourth and final result concerns a basic proposition of Golinskii-Ibragimov
arising in their analysis of the Strong Szeg\"{o} Limit Theorem.
