In this joint work with Zhang Yanyan, we established a theory of delta shock waves with Dirac delta functions developing in both state variables for a class of nonstrictly hyperbolic systems of conservation laws. Firstly, we solved constructively the Riemann problems for the system under consideration. In solutions, we found a kind of delta shock waves on which both state variables simultaneously contain the Dirac delta functions. We strictly proved it satisfies the system in the sense of distributions. The generalized Rankine-Hugoniot relation and entropy condition were proposed to solve the delta shock waves. Secondly, we showed all of the existence and stability of solutions including the delta shock waves to reasonable viscous perturbations. Thirdly, we exhibited the generality and practicability of our theory on the delta shock waves which can be successfully applied to those systems investigated by Korchinski(1977), Tan, Zhang and Zheng(1991), Ercole(2000), Cheng and Yang(2011), etc. We also gave a simplified approach to solve a 2-D Riemann problem for the system studied by Tan and Zhang(1990) for the 4Js’ case and obtained the explicit formulae of the delta shock waves. Finally, we presented the numerical simulations completely coinciding with theoretical analysis.