  The correlations that give rise to incompressible quantum liquid (IQL) states
in fractional quantum Hall systems are determined by the pseudopotential
$V(\mathcal R)$ describing the interaction of a pair of Fermions in a
degenerate Landau level (LL) as a function of relative pair angular momentum
$\mathcal R$. $V(\mathcal R)$ is known for a number of different Fermion
systems, e.g. electrons in the lowest Landau level (LL0) or the first excited
Landau level (LL1), and for quasiparticles of Laughlin-Jain IQL states.
Laughlin correlations, the avoidance of pair states with the smallest values of
$\mathcal R$, occur only when $V(\mathcal R)$ satisfies certain conditions. We
show that Jain's composite Fermion (CF) picture is valid only if the conditions
necessary for Laughlin correlations are satisfied, and we present a rigorous
justification of the CF picture without the need of introducing an "irrelevant"
mean field energy scale. Electrons in LL1 and quasielectrons in IQL states
(e.g. QEs in CF LL1) do not necessarily support Laughlin correlations.
Numerical diagonalization studies for small systems of Fermions (electrons in
LL0 or in LL1, and QEs in CF LL1), with the use appropriate pseudopotentials
$V(\mathcal R)$, show clear evidence for different types of correlations. The
relation between LL degeneracy $g=2\ell+1$ and number of Fermions $N$ at which
IQL states are found is known for a limited range of $N$ values. However, no
simple intuitive models that we have tried satisfactorily describe all of the
systems we have studied. Successes and shortcomings of some simple models are
discussed, and suggestions for further investigation are made.
