The interaction of the solar wind with the Very Local Interstellar Medium is affected significantly by collisional charge exchange of interstellar neutral atomic hydrogen and protons in the heliosheath. This interaction provides an additional net momentum toward the Sun, decreasing the size of the heliospheric cavity. The interaction is complicated by time variations in the internal solar wind flow, the presence of both the interstellar and interplanetary magnetic fields, and the large mean free paths for charge exchange. Proper treatment of the problem calls for a fully six‐dimensional, time‐dependent kinetic interaction model, yet computational complexities inherent in such a model have precluded its full implementation. The role of neutral hydrogen and charge exchange has had a long history beginning with Axford et al.. Refinements in measurements and inferences of properties of the interstellar medium have narrowed the relevant parameter space, as the evolving power of computers has made more sophisticated numerical models possible. Nonetheless, fully numerical models remain out of reach. At the same time, the Voyager spacecraft continue their journey to the region where the neutral interactions are important, giving rise to the need for better and better models. While the ion populations can be approximated to first order by convected Maxwellians, neglecting non‐Maxwellian features associated with pickup, the neutral population acquires non‐negligible non‐Maxwellian features due to the large mean free paths for charge exchange. Self‐consistent models require the evaluation of the interaction of the two populations, and the models now employed differ primarily in how the neutrals are treated. These treatments broadly split into kinetic and “modified hydrodynamic” groups, the latter being based upon different assumptions of how to handle the Boltzmann collision operator. Kinetic models have typically been two‐dimensional due to limitations on computer time and/or limited in the numerical statistics associated with the simulations. The hydrodynamic‐like approaches have the advantage of allowing more realistic geometries, but are subject to the criticism that they are not “sufficiently kinetic,” raising questions about their accuracy. The history, current status, and limitations of some of these ideas and approaches provide a backdrop and motivation for further progress. © 2004 American Institute of Physics