A line digraph L(D) of a directed multigraph \(D=(V(D),A(D))\) has as its vertex-set being A(D), the set of arcs of D, where (a, b) is an arc of L(D) if and only if there are vertices u, v, and w in D such that \(a=(u,v)\) and \(b=(v,w)\) are in A(D). In this paper, we obtain sufficient and necessary conditions for a line digraph L(D) to be supereulerian and to have a spanning trail, respectively, in terms of certain types of path-cycle covers of D. These results will be applied to show each of the following for a digraph D. (i) If \(|A(D)|\geqslant (|V(D)|-1)^2+1\) , then L(D) is supereulerian. The lower bound on |A(D)| is best possible in the sense that there exists an infinite family of digraphs each of which satisfies \(|A(D)| = (|V(D)|-1)^2\) without a supereulerian line digraph. (ii) There exists a well-characterized digraph family \(\mathcal {M}\) such that if D is strong with \(|A(D)|\leqslant |V(D)|+2\) , then L(D) is supereulerian if and only if \(D\not \in \mathcal {M}\) .
