This doctoral thesis reports analytical and numerical results for the electrostatic response of a dusty plasma to a moving test charge. Two important physical aspects of dusty plasmas, namely grain size distribution and grain charging dynamics were taken into account. It appears that the dynamical charging of the grains in a dusty plasma enhances the shielding of a test charge. To clarify the physics, a separate study is made where the charging is approximated by using a time delay. The resulting potential shows the delayed shielding effect explicitly. was observed that the grain charging dynamics leads to a spatial damping and a phase shift in the potential response. Finally, combining these two physical aspects, generalized results for the electrostatic potential were found incorporating the terms from both grain size distribution and grain charging dynamics. The generalized results contain the previous work where these two effects were studied separately and which can now be found as special limiting cases. This kind of work has relevance both in space and astrophysical plasmas.