  Around 1967, Arveson invented a striking noncommutative generalization of
classical $H^\infty$, known as {\em subdiagonal algebras}, which include a wide
array of examples of interest to operator theorists. Their theory extends that
of the generalized $H^p$ spaces for function algebras from the 1960s, in an
extremely remarkable, complete, and literal fashion, but for reasons that are
`von Neumann algebraic'. Most of the present paper consists of a survey of our
work on Arveson's algebras, and the attendant $H^p$ theory, explaining some of
the main ideas in their proofs, and including some improvements and short-cuts.
The newest results utilize new variants of the noncommutative Szeg\"{o} theorem
for $L^p(M)$, to generalize many of the classical results concerning outer
functions, to the noncommutative $H^p$ context. In doing so we solve several of
the old open problems in the subject.
  We include full proofs, for the most part, of the simpler `antisymmetric
algebra' special case of our results on outers.
