Thermodynamic and structural properties of asymmetric electrolytes in aqueous solution have been studied by means of molecular dynamic and Monte Carlo simulations employing two different short-range potentials and the Ewald summation technique for handling the long-range Coulombic interactions. The macroion carried 20 elementary charges and both monovalent and divalent counterions have been considered. Previous accurate results for the 20:1 soft-sphere system, obtained from integral equations and simulations, have been reexamined in view of the present results. Data from extensive simulations covering a wide concentration range of the 20:1 and 20:2 soft-sphere systems have been obtained. The valency of the counterions did not affect the structure of the system in a qualitative manner. In particular, an effective repulsive potential was operating between the macroions even in the 20:2 system at all concentrations. On the contrary, data from simulations of the 20:1 and 20:2 hard-sphere systems indicated a much stronger dependence on the counterion valency. In the case of divalent counterions, the electrostatic repulsion between the macroions was nearly completely screened, but no attractive component arising from the electrostatic interaction could be discerned. Hence, the type of the short-range part of the macroion-counterion potential plays a decisive effect on the long-range structure of the system. Finally, we also discussed the optimization of the Ewald summation and the choice of dielectric boundary conditions applied to asymmetric electrolytes.