  Let $H$ be a $k$-uniform $D$-regular simple hypergraph on $N$ vertices. Based
on an analysis of the R\"odl nibble, Alon, Kim and Spencer (1997) proved that
if $k \ge 3$, then $H$ contains a matching covering all but at most
$ND^{-1/(k-1)+o(1)}$ vertices, and asked whether this bound is tight. In this
paper we improve their bound by showing that for all $k > 3$, $H$ contains a
matching covering all but at most $ND^{-1/(k-1)-\eta}$ vertices for some $\eta
= \Theta(k^{-3}) > 0$, when $N$ and $D$ are sufficiently large. Our approach
consists of showing that the R\"odl nibble process not only constructs a large
matching but it also produces many well-distributed `augmenting stars' which
can then be used to significantly improve the matching constructed by the
R\"odl nibble process.
  Based on this, we also improve the results of Kostochka and R\"odl (1998) and
Vu (2000) on the size of matchings in almost regular hypergraphs with small
codegree. As a consequence, we improve the best known bounds on the size of
large matchings in combinatorial designs with general parameters. Finally, we
improve the bounds of Molloy and Reed (2000) on the chromatic index of
hypergraphs with small codegree (which can be applied to improve the best known
bounds on the chromatic index of Steiner triple systems and more general
designs).
