Leaders of the group of the world's seven wealthiest countries, with Russia making up the eighth, will have different demands on President Bush during the summit in Genoa.
President Vladimir Putin, as leader of the world's second-largest nuclear power, will concentrate on NATO expansion and President Bush's missile defence programme. He will remind Mr Bush that the 1972 Anti-Ballistic Missile Treaty (ABM) is the cornerstone of world security and tell him that if he breaches it a whole series of pacts against nuclear proliferation may unravel.
Mr Putin may agree to allow changes in ABM to permit Mr Bush's missile programme to go ahead. Part of this could be a guarantee, in private, that NATO will not expand to Latvia, Lithuania and Estonia.
Mr Bush's European allies, supported by Canada, will argue against a unilateral approach to major issues. Treaties are there, they will contend, to be kept or renegotiated rather than to be broken. If you renege on a treaty, will anyone want to go into an agreement with you in future?
All the other leaders will press for details of Mr Bush's proposed alternative to the Kyoto Protocol on global warming, with EU members insisting that Kyoto must be implemented and Japan and Canada prepared to make changes.
As for the vast majority of the world's population, which will not be represented in Genoa, the cancellation of Third World debt and urgent action on the AIDS epidemic will top the agenda.