That is a question being asked at Westminster, in political and boardroom meetings, and at official functions across the capital. The idea of the job has generated much excitement, yet it does not even exist.
Although London has its fair share of mayors, including a Lord Mayor of London appointed from among the aldermen of the City, a Lord Mayor of Westminster, and mayors of its other 31 boroughs, they are all honorary appointments, and no one person represents the capital.
Now, an increasing number of people are saying that London should have its own mayor - but somebody with the clout that comes from a democratic mandate.
In three weeks' time, Tony Blair will again publicly give his personal backing to the idea of a mayor for London - a view that has put him at odds with his front-bench environment spokesman, Frank Dobson, who is completely opposed to the suggestion.
A year ago Mr Blair asked Mr Dobson, who is responsible for Labour local- government policy, to prepare a document detailing proposals for a strategic authority for London. The Dobson paper has been a long time coming, but is now due out in a few days. But it will go no further than merely mentioning an elected mayor.
In a speech in London last month, Mr Blair declared: "Though this is controversial and is not yet party policy, I also favour directly-elected mayors, at least for our capital city."
According to the Labour leader, the experience of Paris and New York make it clear what an effective force an elected mayor would be; that it was "one way of restoring civic pride and giving new direction to local communities".
If directly-elected mayors were introduced, a shake-up of local-government power and influence would follow, which is one reason why Mr Dobson's intransigence does not surprise Tony Travers of the London School of Economics, an expert on the government of London.
He said: "An elected mayor of London would be one of the most interesting jobs in Britain, but it would be such a high-profile figure that it would threaten the power of existing councillors. Nevertheless, it would be good for democracy. Too many people think local government boring. This would make it fun."
This week, representatives of London's town halls will meet to discuss the need for a London-wide elected authority, 10 years after the demise of the Greater London Council. The need for a mayor is up for debate, but some London councillors think it a red herring.
Others share Mr Blair's view, that one directly-elected mayor would give the capital a much- needed personal face, and help co-ordinate the city as a whole.
At one lunchtime meeting at the Savoy hotel last month, businessmen agreed that it was time London had a mayor of its own. These were the very same people who cracked open the champagne when the GLC closed its doors for good.
Now they agree that London has suffered by being the only important capital in the West without an elected body with overall authority to run it. Instead, London is run from Millbank Tower by the Government Office for London, as well as the boroughs and a baffling group of quangos carrying out the former functions of the GLC.
Stuart Lipton, one of the developers who created the Broadgate complex in the City, one of London's most recent landmarks, said: "It is quite clear that London needs somebody to be in charge, somebody with flair and invention, but there has to be a proper framework for running the city."
If London followed the example of Paris, the mayor of London would be a name off a party list. The New York version is an official with executive powers.
But what London may well get is someone who combines the role of political leader with the ceremonial role of mayor - someone akin to Joseph Chamberlain when he was both leader and mayor of Birmingham.
The mayoralty would then act as the focus of strategic planning for the capital, and as the voice of London, meaning trouble for all sorts of people, including the secretaries of state for the environment, transport, education, industry, health and heritage, used to working in a system of extraordinarily centralised government.
But such a figure, according to Max Hastings, editor of the London Evening Standard, is essential.
"It would be inspirational, someone who would add glamour," he said.
"I don't want to see the return of a bureaucracy like the GLC, or someone who is a party hack, but someone who would speak up for London."
Neil Kinnock made much the same pledge in 1992 - that Labour would "give Londoners the right to elect a new Greater London authority".Reuse content