  Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$
satisfying $\delta(G) \geq n/2$ contains a perfect matching. We generalize this
to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex
$k$-uniform linear hypergraph $H$ with minimum degree at least $\frac{n}{k} +
\Omega(1)$ contains a matching that covers at least $(1-o(1))n$ vertices. This
minimum degree condition is asymptotically tight and obtaining perfect matching
is impossible with any degree condition. Furthermore, we show that if
$\delta(H) \geq (\frac{1}{k}+o(1))n$, then $H$ contains almost spanning linear
cycles, almost spanning hypertrees with $o(n)$ leaves, and ``long
subdivisions'' of any $o(\sqrt{n})$-vertex graphs.
