We show the following results regarding complete sets:

NP-complete sets and PSPACE-complete sets are many-one
autoreducible.

Complete sets of any level of PH, MODPH, or
the Boolean hierarchy over NP are many-one autoreducible.

EXP-complete sets are many-one mitotic.

NEXP-complete sets are weakly many-one mitotic.

PSPACE-complete sets are weakly Turing-mitotic.

If one-way permutations and quick pseudo-random generators exist,
then NP-complete languages are m-mitotic.

If there is a tally language in NP \cap coNP - P, then, for
every \epsilon > 0,
NP-complete sets are not 2^{n(1+\epsilon)}-immune.

These results solve several of the open questions raised by Buhrman and
Torenvliet in their 1994 survey paper on the
structure of complete sets.