First steps toward a broad climate agreement, such as the Kyoto Protocol, have focused attention on agreement with less than global geographic coverage. We consider instead a policy that is less comprehensive in term of greenhouse gases (GHGs), including only the non-CO2 GHGs, but is geographically comprehensive. Abating non-CO2 GHGs may be seen as less of a threat to economic development and therefore it may be possible to involve developing countries in such a policy who have thus far resisted limits on CO2 emissions. The policy we consider involves a GHG price of about $15 per ton carbon-equivalent (tce) levied only on the non-CO2 GHGs and held at that level through the century. We estimate that such a policy would reduce the global mean surface temperature in 2100 by about 0.57 degrees C; application of this policy to methane alone would achieve a reduction of 0.3 to 0.4 degrees C. We estimate the Kyoto Protocol in its current form would achieve a 0.30 degrees C reduction in 2100 if all Annex B Parties except the US maintained it as is through the century. Furthermore, we estimate the costs of the non-CO2 policies to be a small fraction of the Kyoto restriction. Whether as a next step to expand the Kyoto Protocol, or as a separate initiative running parallel to it, the world could make substantial progress on limiting climate change by pursuing an agreement to abate the non-CO2 GHGs. The results suggest that it would be useful to proceed on global abatement of non-CO2 GHGs so that lack of progress on negotiations to limit CO2 does not allow these abatement opportunities to slip away.Abstract in HTML and technical report in PDF available on the Massachusetts Institute of Technology Joint Program on the Science and Policy of Global Change website (http://mit.edu/globalchange/www/).