Let G = (V, E) be a strongly connected directed graph. The edge connectivity λ of G is the minimum number of edges whose deletion leaves a graph that is not strongly connected. Computing the edge connectivity of a graph is a classical subject in graph theory, and is an important notion in several application areas, such as transportation, communication, production, scheduling, and power engineering. In this paper we explore the design space of efficient algorithms for computing the edge connectivity of a directed graph in practice. In particular, we present efficient implementations of Gabow's algorithm, which is based on matroid intersection and packing spanning trees, as well as algorithms based on recent “local search” algorithms for minimum-cut. We conduct a thorough empirical study to highlight the merits and weaknesses of each technique.