  We investigate space curves with large cohomology. To this end we introduce
curves of subextremal type. This class includes all subextremal curves. Based
on geometric and numerical characterizations of curves of subextremal type, we
show that, if the cohomology is ``not too small,'' then they can be
parameterized by the union of two generically smooth irreducible families; one
of them corresponds to the subextremal curves. For curves of negative genus,
the general curve of each of these families is also a smooth point of the
support of an irreducible component of the Hilbert scheme. The two components
have the same (large) dimension and meet in a subscheme of codimension one.
