  We prove that for every fixed $k$, the number of occurrences of the
transitive tournament $Tr_k$ of order $k$ in a tournament $T_n$ on $n$ vertices
is asymptotically minimized when $T_n$ is random. In the opposite direction, we
show that any sequence of tournaments $\{T_n\}$ achieving this minimum for any
fixed $k\geq 4$ is necessarily quasi-random. We present several other
characterizations of quasi-random tournaments nicely complementing previously
known results and relatively easily following from our proof techniques.
