  A conjecture of Thomassen from 1982 states that for every k there is an f(k)
so that every strongly f(k)-connected tournament contains k edge-disjoint
Hamilton cycles. A classical theorem of Camion, that every strongly connected
tournament contains a Hamilton cycle, implies that f(1)=1. So far, even the
existence of f(2) was open. In this paper, we prove Thomassen's conjecture by
showing that f(k)=O(k^2*log^2(k)). This is best possible up to the logarithmic
factor. As a tool, we show that every strongly 10^4*k*log(k)-connected
tournament is k-linked (which improves a previous exponential bound). The proof
of the latter is based on a fundamental result of Ajtai, Koml\'os and
Szemer\'edi on asymptotically optimal sorting networks.
