They are supposed to decide on a new President of the Commission to succeed Jacques Delors. The Belgian Prime Minister, Jean- Luc Dehaene, is the favourite, with the backing of most member states. But Ruud Lubbers, the Dutch Prime Minister and the other main candidate, seems unwilling to back down. If he does not - if a mixture of national pride and personal psychology prevents a decision - there will be deadlock and another summit will have to follow.
EU electors have just voted for a new European Parliament. But the choice of a President of the Commission is one of the least democratic processes in the European Union, with no input from outside the circle of heads of government.
Mr Dehaene is the leading candidate for one reason, which is that Helmut Kohl, the German Chancellor, likes him. In addition, Francois Mitterrand, the French Prime Minister, is glad to see a candidate who speaks French. In general the other states follow the lead of these two. But any of those gathered around the dinner table tonight can veto, because each is a sovereign head of government.
The sounds of the Trooping of the Colour wafted incongruously through the press room in Corfu yesterday, courtesy of the BBC and cable television, the real sounds of antique sovereignty. European summits, too, have something of this ritual quality: every six months a different country holds the Presidency of the European Union, the heads of government meet in that country, photos are taken and gifts are given. The Palace of St Michael and St George, where the Corfu summit will be held, exudes this quality of ritual.
This is all about the parading of national status. The country that runs the EU tries to make as good a fist of its summit as possible, pushing its view of Europe and its style of doing business. The Greeks have done this in an often hamfisted way, running into trouble over Macedonia and alienating their colleagues in the process. It is enough to make anyone think twice about the system of allowing each country to run the EU for six months.
But the summits are also a key part of the EU's decision-making machinery, where the really tough deals get made, or not as the case may be. Massive new infrastructure projects will also be planned here, for instance. The way this happens is bound up in national deal-making, trades and influence- peddling. It is hardly very effective and not always very pretty, but that is the way world leaders work.
The question of the EU's functioning will also feature on the agenda today. In a little over 18 months the EU has to assemble again in an intergovernmental conference (IGC), a meeting that will rewrite the Union's rules and finish in a summit that will be the heir of Maastricht. The EU will set up a committee before then to consider the issues, and the French, German and Spanish governments, who will run the EU for the next 18 months, have pledged to co-ordinate the run-up to 1996.
The issues that the intergovernmental conference considers will be very much to do with the way that this ill-fitting system works. The six-monthly presidency system will be revised; the Commission may be given more power to run things at supranational level in some well-defined areas, and the complex voting systems considered. The more states that come in the more difficult it becomes to get decisions made in the EU. In the distance, there are six central European states queueing up. Can national vetoes be sustained with 22 members? Can small nations continue to have as much weight as large ones? Can every nation have a Commissioner?
An IGC, too, is an undemocratic process, carried out behind closed doors by government representatives. One of the issues the EU leaders will consider today is whether the European Parliament ought to be drawn into this charmed circle. The Parliament was a beneficiary of Maastricht. It will, for instance, have the right to withold its approval of Mr Delors' successor, though not to choose him.
But the European Parliament, as voters showed when they went to the polls earlier this month, is not held in high esteem by Europe's electorates. Its powers are restricted, it is a toy compared to national legislatures, and it functions poorly. The EU heads of government are likely to keep the Parliament on the sidelines, just as they will fight to retain their national freedom to veto when they rewrite the rules in 1996.
What this is all about is not just democracy, but also legitimacy: whether or not the EU is seen and felt to have rightful authority. Legitimacy is still, in the main, rooted in national governments and nation-states, in France, Denmark, and Britain in particular. The arcane system for choosing presidents is a constant reminder to the Commission that however uppity it may become, it is the creation of national governments.
Jacques Delors has tried to create a different sense of legitimacy, one rooted in the concrete benefits of integration. Though as much the creature of national choice as any other Commission president, he made the job something more. And he focused on creating greater prosperity for citizens as the way to gain their allegiance, through the single market and projects like the new infrastructure links.
It worked to some extent; Mr Delors leaves the post vastly enhanced. But the EU's authority has also been gravely weakened in the last few years, as national aspirations have resurfaced and integration has flagged.
That is why meetings of the leaders of those nation-states still have such importance, and that is why the pageantry, the posing and the photo-opportunities still matter. It is the only way that the EU can work, at the moment; but it is not a particularly effective way to run a railroad, let alone a Trans- European Network.