By Bridget Kendall
BBC diplomatic correspondent
Obama returns home after a week of successful speech-making in Europe
So Barack Obama is back home after his eight-day whirlwind odyssey - his first proper foray overseas as US President.
Time for him to snatch a family holiday with his girls and prepare for that new White House puppy there has been so much media chatter about.
Time for the rest of the world to mull over what has been learnt about his presidential priorities and promises for tackling global challenges.
There is no doubt that the contours of his foreign policy are taking shape. A scattergun of speeches, high on rhetoric and rich in crafted messages, have targeted specific audiences.
And all, it seems, have been won over: enthusiastic Brits, excited continentals, gratified Turks, and weary, gritty US troops, still counting the days of their Iraq combat duty.
How much distance can Obama put between his and Bush's policies?
The range of his remarks has been impressive, the tone supple and carefully calibrated. Apart from one evening news conference in London where jet lag and a cold seemingly caught up with him, he was fluent and inspirational.
He marked clear blue water between himself and his predecessor. He admitted America had been wrong on some things and would change course if it made new errors.
This was a refreshingly humble tune to the ears of foreigners who had been infuriated by what they heard as a stubborn drumbeat of unilateralism from the previous president.
He did not dodge the awkward questions. Even the touchy issue of genocide which still enflames Turkish-Armenian relations was elegantly, though indirectly, dealt with. Without upsetting his Turkish hosts by repeating his campaign pledge for Armenia's grievance to be recognised, he delicately urged the two sides to focus instead on their mutual future.
Heady though his rhetoric may be, it cannot entirely conceal curious inconsistencies and the shadow of future difficulties.
Take the perennially controversial question of America's global leadership.
"We have come to listen as well as lead
We may not always have the best answer," he and his officials repeated endlessly.
Perhaps it was different behind closed doors, but in public his folksy "town hall meetings" and press conferences were, above all, an opportunity for him to do the talking. It was his audience who did most of the listening.
Falling into step
His cadences reinforced the impression of a preacher, ready to inspire and guide a wandering flock. "The challenge is great... so many have lost so much," he intoned, an orator up high upon a podium - even if he did deliberately deflate the balloon of his own authority now and again with conversational humour. "I think we did OK," he replied at one point about the G20 summit, with disarming simplicity.
Mr Obama was the undisputed centre of attention at recent summits
The point is that Barack Obama still wants to proclaim the fact of American exceptionalism.
"America is a critical actor on the world stage and we should not be embarrassed about that," he declared.
But the further point is that, in his case, America's allies are still ready to fall in step behind him
for several reasons.
He likes to argue that it is because his unlikely ascendancy to the presidency is an embodiment of his political message that anything is possible.
But he is also quite simply a global celebrity, a political rock star - the undisputed centre of attention at all the many summits he attended.
Recall only the exuberant shout of "Mr Obama" by Silvio Berlusconi at the Queen's photo shoot - so loud that it earned the Italian Prime minister a royal reprimand.
Remember the cat-like grin on the face of Gordon Brown as he basked in President Obama's praise of his London summit-hosting skills and the reassurance that Britain still somehow merits a "special relationship".
Mantle of leadership
Mr Obama appears to stand head and shoulders above other leaders
And look closely at that picture of the Obamas posing for cameras next to the French presidential pair in Strasbourg. Nicholas Sarkozy stands on anxious tiptoe next to his willowy wife, but is utterly dwarfed by the towering figures of Michelle and Barack Obama. It had a symbolic resonance, as though European leaders preoccupied with internal rivalries and their own self importance, lack the stature to see President Obama's further horizons.
And that, perhaps, is the third reason why President Obama can get away with claiming that the US still deserves the mantle of global leadership. His vision is bold and his mission, he tells us, is to galvanize international collective action to solve not just problems of the next four or eight years, but of future generations.
George W Bush fought a war on terror. Terrorism remains one of Barack Obama's scourges too, but so does global warming and a newly revived nightmare of Armageddon from misplaced nuclear weapons.
No longer does the US President invoke a fight for freedom and democracy as the cure for the world's ills, as George W Bush did. Barack Obama's call is to liberate humankind from fear, of the cataclysmic natural and man-made disasters that may overwhelm us if we don't act together.
But a few niggling loose ends mean there are a few contradictions here.
For all talk of change, policy on Iraq and Afghanistan is familiar
Firstly, yes, the apocalyptic vision may be different. But examine policy on a day-to-day level, and the dividing lines between the old Bush and new Obama administration look decidedly smudgy. For all Obama's talk of change, the general direction of American foreign policy is surely rather familiar.
In both Iraq and Afghanistan, the exit strategies rest on training up the local army and police to do the security job themselves.
On Middle East Peace, the aim remains to get the Israelis and Palestinians to talk about a two-state solution, with no indication yet that the White House might rethink its refusal to talk to Hamas or Hezbollah militants.
On Iran the policy is still carrot and stick, though the carrot has been sweetened slightly by the inclusion of Americans in the official negotiating team.
And even President Obama's much vaunted reset button in relations with Russia has broken no real new ground yet- either on missile shields or Nato expansion. An offer to sit down with Russia to discuss new cuts in nuclear arsenals is exactly where Presidents Bush and Putin started out - and look how that love affair ended.
Secondly, how does one join up the dots between Obama the visionary, whose goal is to rid the world of nuclear weapons and other threats, and Obama the pragmatist, who sees the way to do this is to use US leadership to "guide a process of orderly integration"?
One goal is to repair relations with Russia and China
If his philosophy is to engage with global leaders across the board, how far is he prepared to go? Is he still willing to confront nations about human rights abuses or other worrying behaviour?
Or has the bigger strategic goal of repairing relations with Russia and China eclipsed the human rights agenda, so that from now on issues like Russia's still incomplete withdrawal from Georgian territory, or China's treatment of protesters in Tibet will be put firmly on the backburner?
Is Obamaland a return to realpolitik and an end to alliances built on values?
And there is a third area of apparent contradiction.
On the one hand President Obama sends a message of flexibility, an appreciation of complexity. But he sometimes comes across as a leader who, when pushed, will put his foot down.
In Strasbourg, addressing young Europeans, Barack Obama declared that the fight to keep al-Qaeda at bay did not need to lead to a compromising of moral values. This was why he was closing the Guantanamo Bay detention centre, he said, and outlawing the use of torture in interrogations.
In Prague he argued that the rules to prevent transgressions against nuclear agreements must be binding, and "violations must be punished."
In Ankara, indicating his support for the Turkish government's fight against the Kurdish militant movement PKK, he announced "there is not excuse for terror against any nation."
It sounds so principled and categorical: an American President prepared to send tough messages and take decisive action.
Not all nations want to be guided into a 'process of orderly integration'
And he has already shown there is a hint of a streak of ruthlessness in him.
Take his determination to keep going with US airstrikes on suspected al-Qaeda hideouts in Pakistan's border areas, despite the risk of civilian deaths and government protests.
Take his readiness at home to contemplate bankruptcy for giants of the US car industry.
And take the steel in his voice when he addressed European leaders about his expectation that they would step up to the plate to do their bit when it came to more resources for Afghanistan.
At the moment he can do no wrong, but once the honeymoon is over and Obamamania subsides, there is plenty of room here for tensions and resentments.
North Korea for one appears deaf to Obama's rhetoric
But the final twist is that the Obama vision of a world that is willing to pull together, and the Obama method of using the weapon of rhetoric and persuasion to win over converts, has already run into the brick wall of reality.
Not all leaders are susceptible to the Obama magic. Not all nations want to be guided into a "process of orderly integration".
Only hours before his nuclear speech in Prague, the North Koreans made that abundantly clear. They defied American and global appeals and went ahead with their rocket test launch, threatening to raise the stakes still further if UN sanctions were tightened.
And in response, it turns out that not much has changed at the UN Security Council either. President Obama may speak winningly and the handshakes from last week's summits may paint a rosy picture of collaboration, but Russia and China have not budged from their recent reluctance to endorse more UN sanctions.
Business as usual.
An early reminder that it will take more than fine speeches to reshape the world. And once the low hanging fruit of easy diplomatic gains have been plucked, the course of American foreign policy may be just as rocky as it has ever been.
You can listen to the BBC World Service's The Forum with Bridget Kendall