properties give rise to the same notion of quasirandomness. Before stating some of their results we require a little notation. A tournament $T = (V,E)$ consists of a set of vertices $V  = V(T)$, together with a set of edges $E  = E(T) \subset V \times V$, with the property that (i) $(u,u) \notin E$ for all $u\in V$ and (ii) exactly one of $(u,v)$ or $(v,u)$ lies in $E$ for all distinct $u,v \in V$. We often write $\overrightarrow {uv}$ to denote an edge $(u,v)$. A tournament $H$ appears as a subtournament of $T$ if there is a map $\phi: V(H) \to V(T)$ such that $\overrightarrow {uv} \in E(H)$ if and only if $\overrightarrow {\phi (u)\phi (v)} \in E(T)$. The map $\phi $ is said to be a labelled embedding of $H$ into $T$. Let $$N^*_T(H)  
            = 
        \big  | \big  \{ \phi : V(H) \to V(T): 
        \phi \mbox{ is a labelled embedding of } H \mbox { into } T \big  \} \big |.$$ Given $U \subset V(T)$ let $T[U]$ denote the subtournament of $T$ induced by the vertex set $U$. Also let $N^*_T(H;U) := N^*_{T[U]}(H)$. For $u \in V(T)$ let $d^+_T(u) = |\{v\in V(T): \overrightarrow{uv} \in E(T)\}|$ and 