Needed reals and recursion in generic reals

We consider sets of reals that are "adequate" in various senses, for exampl e dominating or unbounded or splitting or non-meager. Call a real x "needed " (in any of these senses) if every adequate set contains a real in which x is recursive. We characterize the needed reals for numerous senses of "ade quate." We also consider, for various notions of forcing that add reals, th e problem of characterizing the ground-model reals that are recursive in ge neric reals. (C) 2001 Elsevier Science B.V. All rights reserved.