As President Bush returned to the US from a fence-mending tour of Europe, the American media said he was largely successful in improving relations.
US media said Mr Bush succeeded in closing rifts with Europe
But the press recognised that there was vast improvement after transatlantic relations broke down in the run-up to the war in Iraq.
However, while the charm offensive was deemed a success, the American media saw a number of challenges ahead, including the EU's plans to sell arms to China and relations with Russia.
NBC's Tim Russert said that the dialogue between the US and Europe had reopened: "Once people are talking, a whole lot of things get easier."
But President Bush and other countries had to strike a delicate balance with Russian President Vladimir Putin, Mr Russert said.
Mr Bush has enjoyed a very cordial relationship with President Putin, but at the same time, he is facing pressure from some in Congress, including Republican Senator John McCain, to press Mr Putin to end the erosion of democracy in Russia.
"The world needs [Russia's] help in fighting terrorism. The world needs its help in gathering nuclear material that's left over from the Soviet days," Mr Russert said.
"But at the same time, the world has to see some progress on the democratic front."
On FoxNews, PJ Crowley, former special assistant to President Clinton, said: "Privately, I was hoping the president would take a sharper line" with Mr Putin.
Some in the US wanted more pressure on Mr Putin to reform
"He [Bush] clearly went to bat for democracy yesterday, [but] I thought he swung and missed," Mr Crowley said.
But Republican strategist Rich Galen countered on Fox that President Bush probably took a much tougher line with the Russian president behind closed doors than he did in their press conference.
"Everybody who follows this that's being fair about it understands that what happened in private was probably pretty severe and that what President Bush was doing was letting President Putin have a little bit of breathing room for home consumption," Mr Galen said.
But the US media and many members of Congress were pleased that Mr Bush and Mr Putin were able to agree on improved security at Russian nuclear sites and the promises to curb the sales of shoulder-fired missiles.
"It would be naive to take at face value comments President Bush and Russia's President Vladimir Putin made yesterday after their meeting in the Slovakian capital, Bratislava," an editorial in the Boston Globe said.
But "their agreement to improve the security of nuclear sites and materials heralds an important practical measure that benefits everyone but the terrorists targeting Americans and Russians", it added.
Chill over China
The American media also said that the European Union's plans to lift the arms embargo on China represented a major policy challenge to the US.
The US is openly concerned about an ascendant China and worries that the arms sales will help the rapid economic and military expansion of the rising Asian giant.
"In an overt show of teeth, Bush expressed 'deep concern' over Europe's plan to lift the arms embargo against China, warning that any action that could threaten Taiwan might enrage the US Congress," wrote Dan Froomkin of the Washington Post.
Mark Landler of the New York Times wrote: "The conflict has bubbled up during President Bush's visit here this week, injecting a discordant note into his otherwise harmonious tour."
"Lifting the embargo has become a symbol of the European Union's efforts to deepen its relationship with China, which it views as a strategic partner rather than merely an ally," Mr Landler wrote.