This is the first US presidential election since 1928 in which neither the president nor the vice-president is standing in the primaries, seeking re-election.
By Justin Webb
BBC North America editor
It is the Nobody Knows election. Nobody knows which candidates will end up representing the two main parties, nobody knows exactly when the parties will choose them, and nobody knows which issues will decide the eventual contest.
Mr Giuliani (right) can afford to lose early primaries and focus on Florida
We are looking at two elections, each with its own dynamic, the nomination battle - the primaries and caucuses - and the general election itself.
The landscape often changes dramatically during the course of the battle.
John Kerry was not the Democratic frontrunner before the Iowa caucus in 2004. George W Bush was nearly derailed by the New Hampshire primary in 2000.
Two candidates bashing each other can allow a third to slip through and capture the affection of the voters.
The first hurdle
And the voters enjoy their power.
A recent poll suggested half of them in New Hampshire are not intending to make up their minds until the week before the contest on 8 January.
Both Rudy Giuliani and Hillary Clinton have been waging national campaigns which might blow up in their faces.
EARLY PRIMARY TIMELINE
3 Jan - Iowa caucus
5 Jan - Wyoming caucus (Rep)
8 Jan - New Hampshire
15 Jan - Michigan
19 Jan - Nevada caucus, and South Carolina primary (Rep)
26 Jan - South Carolina primary (Dem)
29 Jan - Florida primary
Rudy Giuliani has remained true-(ish) to his previously held conviction that abortion should be reasonably widely available to American women. This might hurt him among conservative voters in rural Iowa, but it is close to the view of most Americans as expressed in national polls.
Hillary Clinton has voted in favour of sanctions on Iran demanded by the White House. Among her party's left wing anti-war supporters in the early primary states this caused dismay, but in the wider population it is probably the position the majority would back.
What would happen if either or both candidates came unstuck in Iowa and New Hampshire?
The traditional view is that they would crash and burn - like Howard Dean in 2004 - and we would hardly remember their names by the summer.
But these candidates might be different; they have national recognition, national campaign staff and national funds.
So Mr Giuliani could perhaps brush off early losses and focus on Florida which comes at the end of January and which he should win.
A third candidate?
Hillary Clinton could do the same and focus on Super Tuesday in the first week of February where New York, New Jersey, and California will be among the big states she will expect to win.
Mike Bloomberg could spend $1bn on a campaign and not miss it
There is also less time this year between Iowa and Super Tuesday - less time for momentum to be built for an Iowa winner or lost for a loser.
The candidates from each of the main parties are likely to be anointed after Super Tuesday or within a month or so of that contest.
So by March it is very likely that a Democrat and a Republican will have emerged.
It is also possible, particularly if large segments of the middle ground in US politics are unhappy (say, with a choice between John Edwards and Mike Huckabee) that the Mayor of New York, Mike Bloomberg, might step in.
Mr Bloomberg does not need to raise any money. He could spend $1bn (£500m) on the campaign and not miss it.
Although he was technically a Republican until recently, his gut sympathies are with the Democrats and he would attract large numbers of centre-left voters.
He is a serious politician (do not think Ross Perot) and he will only stand if he could win.
Iraq and the economy
His entry - perhaps as late as March - would focus on technical competence, leadership skills, and business acumen, allied with social awareness and concern.
It would be the ultimate "throw out the party hacks" candidacy and there is no doubt that polling in recent months suggests it strikes a chord with large numbers of Americans.
As for the issues in the second contest, the general election in November, Iraq is plainly important but is it necessarily damaging for the Republicans?
In a nation of 300 million people, bristling with military power and at war around the world, managing an orderly transition of power is a majestic enterprise
Might an Iraq narrative emerge that makes "staying the course" look statesmanlike and wise?
The economy likewise, is plainly an issue, but will it be in catastrophic shape by next November or will it be powering ahead, with house prices on the up again and the market panic of recent times a distant memory?
More than is normal in presidential politics, nobody knows.