  Sumner's universal tournament conjecture states that any tournament on $2n-2$
vertices contains a copy of any directed tree on $n$ vertices. We prove an
asymptotic version of this conjecture, namely that any tournament on
$(2+o(1))n$ vertices contains a copy of any directed tree on $n$ vertices. In
addition, we prove an asymptotically best possible result for trees of bounded
degree, namely that for any fixed $\Delta$, any tournament on $(1+o(1))n$
vertices contains a copy of any directed tree on $n$ vertices with maximum
degree at most $\Delta$.
