  Most of the fractions observed to date belong to the sequences $\nu=n/(2pn\pm
1)$ and $\nu=1-n/(2pn\pm 1)$, $n$ and $p$ integers, understood as the familiar
{\em integral} quantum Hall effect of composite fermions. These sequences fail
to accommodate, however, many fractions such as $\nu=4/11$ and 5/13, discovered
recently in ultra-high mobility samples at very low temperatures. We show that
these "next generation" fractional quantum Hall states are accurately described
as the {\em fractional} quantum Hall effect of composite fermions.
