An (n,k,t,p,q)-directed strongly regular graph is a directed graph with n vertices satisfying (i) each vertex has k out-neighbors and k in-neighbors, including t neighbors counted as both in- and out-neighbors of the vertex; and (ii) the number of paths of length two from a vertex x to another vertex y is p if there is a directed edge from x to y, and is q otherwise. Such graphs were introduced by Duval in 1988 as one of the possible generalization of classical strongly regular graphs to the directed case. In this talk, constructions of directed strongly regular graphs by groups will be given.