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.