Abstract. By L(D) we denote a line digraph of D. We set L⁰(D)=D. For i>0 the i-iterated line digraph of D is Lⁱ(D)=L(L^i-1(D)). We consider three types of eccentricities. Let u be a vertex of D. Then

1. e⁺_D(u)=max{dist_D(u,v) where v is a vertex of D} (out-eccentricity),
2. e^-_D(u)=max{dist_D(v,u) where v is a vertex of D} (in-eccentricity),
3. e_D(u)=max{e⁺_D(u), e^-_D(u)} (eccentricity).

We prove that if D is a digraph for which rad⁺(Lⁱ(D)) is less than infinity for every i, then rad⁺(Lⁱ(D))-i eqals one constant for all i "sufficiently large".

Further, if D is not a cycle then rad(Lⁱ(D))-i is bounded by constants (from both sides).

If D is a strongly connected digraph then there is a superdigraph H of D such that C⁺(Lⁱ(H))=Lⁱ(D) for every i>0. However, if D is not strongly connected, then in some cases there is no digraph H such that C⁺(L(H)) is isomorphic to L(D).

If L^j(D) is not an empty digraph then there is a superdigraph H of D such that C(Lⁱ(H))=Lⁱ(D) for every i< j. However, there are digraphs D for which there is no superdigraph H for which the restriction on i can be removed.