  Tur\'{a}n-type problems have been widely investigated in the context of
undirected simple graphs. Tur\'{a}n problems for paths and cycles in directed
graphs have been treated by Bermond, Heydemann et al. Recently $k$-geodetic
digraphs have received great attention in the study of extremal problems,
particularly in a directed analogue of the degree/girth problem. Ustimenko et
al studied the problem of the largest possible size of a diregular $k$-geodetic
digraph with order $n$ and gave asymptotic estimates. In this paper we consider
the maximal size of a $k$-geodetic digraph with given order with no assumption
of diregularity. We solve this problem for all $k$ and classify the extremal
digraphs. We then provide a partial solution for the more difficult problem of
the largest number of arcs in a strongly-connected $k$-geodetic digraph with
given order and provide lower bounds for these numbers that we conjecture to be
extremal for sufficiently large $n$. We close with some results on generalised
Tur\'{a}n problems for $k$-geodetic digraphs.
