  In this paper, we study degree conditions for the existence of large
matchings in uniform hypergraphs. We prove that for integers $k,l,n$ with $k\ge
3$, $k/2<l<k$, and $n$ large, if $H$ is a $k$-uniform hypergraph on $n$
vertices and $\delta_{l}(H)>{n-l\choose k-l}-{(n-l)-(\lceil n/k
\rceil-2)\choose 2}$, then $H$ has a matching covering all but a constant
number of vertices. When $l=k-2$ and $k\ge 5$, such a matching is near perfect
and our bound on $\delta_l(H)$ is best possible. When $k=3$, with the help of
an absorbing lemma of H\'{a}n, Person, and Schacht, our proof also implies that
$H$ has a perfect matching, a result proved by K\" uhn, Osthus, and Treglown
and, independently, of Kahn.
