Abstract. Dirac’s theorem states that any [Formula: see text]-vertex graph [Formula: see text] with even integer [Formula: see text] satisfying [Formula: see text] contains a perfect matching. We generalize this to [Formula: see text]-uniform linear hypergraphs by proving the following. Any [Formula: see text]-vertex [Formula: see text]-uniform linear hypergraph [Formula: see text] with minimum degree at least [Formula: see text] contains a matching that covers at least [Formula: see text] vertices. This minimum degree condition is asymptotically tight, and obtaining a perfect matching is impossible with any degree condition. Furthermore, we show that if [Formula: see text], then [Formula: see text] contains almost spanning linear cycles, almost spanning hypertrees with [Formula: see text] leaves, and “long subdivisions” of any [Formula: see text]-vertex graphs.