The stress of a vertex in a graph is the number of geodesics passing through it (A. Shimbel, 1953). A graph is $k$-stress regular if stress of each of its vertices is $k$. In this paper, we investigate some results and compute stress of vertices in some standard graphs and give a characterization of graphs with all vertices of zero stress except for one. Also we compute stress of vertices in graphs of diameter 2 and in the corona product $K_m \circ G$. Further we prove that any strongly regular graph is stress regular and characterize $k$-stress regular graphs for $k=0,1,2$.