  A minimum feedback arc set of a directed graph $G$ is a smallest set of arcs
whose removal makes $G$ acyclic. Its cardinality is denoted by $\beta(G)$. We
show that an Eulerian digraph with $n$ vertices and $m$ arcs has $\beta(G) \ge
m^2/2n^2+m/2n$, and this bound is optimal for infinitely many $m, n$. Using
this result we prove that an Eulerian digraph contains a cycle of length at
most $6n^2/m$, and has an Eulerian subgraph with minimum degree at least
$m^2/24n^3$. Both estimates are tight up to a constant factor. Finally,
motivated by a conjecture of Bollob\'as and Scott, we also show how to find
long cycles in Eulerian digraphs.
