  We prove that a tournament with $n$ vertices has more than $0.13n^2(1+o(1))$
edge-disjoint transitive triples. We also prove some results on the existence
of large packings of $k$-vertex transitive tournaments in an $n$-vertex
tournament. Our proofs combine probabilistic arguments and some powerful
packing results due to Wilson and to Frankl and R\"odl.
