Mr Giuliani was elected by the narrowest of margins to improve the quality of life of crime-weary New Yorkers, but it is anybody's guess how things will turn out. Perhaps his greatest asset will be the determination of so many citizens to play their part, through neighbourhood pressure groups, in improving city life.
From his predecessor, David Dinkins, a Democrat and the city's first black mayor, Mr Giuliani inherits a burgeoning budget deficit, an overblown city bureaucracy, and a police force judged by a city commission to be plagued by a culture of corruption, shockingly incompetent and not up to the task of reforming itself.
Elected by middle-class whites fed up with murders, mayhem and the pervasive filth on the city's streets, Mr Giuliani has promised a 'fusion' administration, bringing together all the political, ethnic and religious groups, luring back jobs and preventing an eruption of racial tension.
The mayor of New York must reconcile competing visions. Mr Giuliani has been careful to appoint Jewish, African-American, Hispanic, Asian and female officials to avert criticism, but his immediate battles may be fought on a different plane.
Like other Republican mayors, Mr Giuliani favours reducing government, increasing privatisation, encouraging competition, and putting a greater burden on the poor to take care of themselves.
Mr Giuliani must make cuts in city hall and in services if he is to counter an immediate budget shortfall of dollars 750m ( pounds 500m) and prepare adequately for the projected deficit of dollars 2bn in next year's budget. All this will place immense strain on the new mayor's relationship with the city council and his ability to meet the demands of the city's powerful unions.
In a gracious departure from the Mansion House, Mr Dinkins, blocked efforts by the city council to curb the mayor's powers to privatise city services. The council leadership, which is generally friendly to the unions, had tried to push through a last-minute measure that would have required the mayor to submit detailed cost studies and permit the council to hold hearings on any proposals to shift city enterprises into the private sector. Such a measure could have involved interminable delays, so its defeat is a plus for Mr Giuliani.
On the budget front, Mr Giuliani has already indicated his strategy by asking all city agencies, except for welfare, the schools board and the city university, to prepare plans for budget cuts up to 10 per cent. Even the hard-pressed police department - the largest in the nation at 30,000 - may face funding cuts of 5 per cent.
Mr Giuliani's margin of popularity - he won by 3 percentage points - cannot take many knocks, even small ones. There has been a furious middle-class outcry, for example, over his plan to redistribute city grants to the arts from small theatres and galleries to the bigger museums where the tourists go. Improving city life means attracting tourist dollars plus keeping the neighbourhood theatre going, citizens argue.
In the end, much of a New York mayor's perceived success - and thus his chances of creating an air of bustle conducive to investment and of getting re-elected after four years - depend on his charisma, natural or man-made. John Lindsay, had lots of charisma, and Ed Koch, who ran the city for 12 years, had too much, many would say. David Dinkins had so very little it barely showed, and Mr Giuliani has shown none. So far.