We discuss some problems and conjectures in
higher dimensional operator theory. These all
have something to do with the basic problem of
developing an effective Fredholm theory of d-contractions,
and completing the index theorem that relates the
curvature invariant to the index of a Dirac operator.
These remarks will appear in the "problem book" that
is evolving in electronic form out of a series of
conferences on operator spaces held in CIRM, Luminy,
during the past several years.