  In this expository paper we describe the study of certain non-self-adjoint
operator algebras, the Hardy algebras, and their representation theory. We view
these algebras as algebras of (operator valued) functions on their spaces of
representations. We will show that these spaces of representations can be
parameterized as unit balls of certain $W^{*}$-correspondences and the
functions can be viewed as Schur class operator functions on these balls. We
will provide evidence to show that the elements in these (non commutative)
Hardy algebras behave very much like bounded analytic functions and the study
of these algebras should be viewed as noncommutative function theory.
