  A subgraph of an edge-coloured graph is called rainbow if all its edges have
different colours. The problem of finding rainbow subgraphs goes back to the
work of Euler on transversals in Latin squares and was extensively studied
since then. In this paper we consider two related questions concerning rainbow
subgraphs of complete, edge-coloured graphs and digraphs. In the first part, we
show that every properly edge-coloured complete directed graph contains a
directed rainbow cycle of length $n-O(n^{4/5})$. This is motivated by an old
problem of Hahn and improves a result of Gyarfas and Sarkozy. In the second
part, we show that any tree $T$ on $n$ vertices with maximum degree
$\Delta_T\leq \beta n/\log n$ has a rainbow embedding into a properly
edge-coloured $K_n$ provided that every colour appears at most $\alpha n$ times
and $\alpha, \beta$ are sufficiently small constants.
