Recent computational and theoretical progress in understanding and calculating ion collection by negatively charged absorbing objects in a flowing plasma is outlined. The results are placed in the context of key theoretical achievements of prior research. Despite the topic's long history, and past profound insights, fully rigorous quantitative solution of the non-linear, multidimensional, self-consistent, kinetic-theory problem has not until recently been feasible. Now we are able to establish the adequacy or inadequacy of approximate treatments, and provide critical quantitative results. In the process, some qualitative surprises have also emerged.