We extend the classical Helson-Szego theorem to the context of noncommutative Hardy spaces constructed from maximal subdiagonal subalgebras of finite von Neumann algebras. We then briefly survey the existent theory of Toeplitz Operators on these spaces, before going on to show how this noncommutative Helson-Szego theorem may be used to characterise invertibility of these operators. These results were obtained jointly with Quanhua Xu, and complement and extend classical results of A Devinatz and HR Pousson.