  We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$,
$d\geq1$, for cubic, quintic, focusing and defocusing interactions. For both
the focusing and defocusing case, and any $d\geq1$, we prove local existence
and uniqueness of solutions in certain Sobolev type spaces $\cH_\xi^\alpha$ of
sequences of marginal density matrices. The regularity is accounted for by
$\alpha>\frac12& if $d=1$, $\alpha>\frac d2-\frac{1}{2(p-1)} $ if $d\geq2$ and
$(d,p)\neq(3,2)$, and $\alpha\geq1$ if $(d,p)=(3,2)$, where $p=2$ for the
cubic, and $p=4$ for the quintic GP hierarchy; the parameter $\xi>0$ is
arbitrary and determines the energy scale of the problem. This result includes
the proof of an a priori spacetime bound conjectured by Klainerman and Machedon
for the cubic GP hierarchy in $d=3$. In the defocusing case, we prove the
existence and uniqueness of solutions globally in time for the cubic GP
hierarchy for $1\leq d\leq3$, and of the quintic GP hierarchy for $1\leq d\leq
2$, in an appropriate space of Sobolev type, and under the assumption of an a
priori energy bound. For the focusing GP hierarchies, we prove lower bounds on
the blowup rate. Also pseudoconformal invariance is established in the cases
corresponding to $L^2$ criticality, both in the focusing and defocusing
context. All of these results hold without the assumption of factorized initial
conditions.
