Moscow: Champagne corks popped in Moscow as Russians celebrated the start of Donald Trump's presidency, while there were guarded congratulations from Mexico and fears of a trade war in Berlin.
Mexico's President Enrique Pena Nieto said on Twitter that while the sovereignty, national interest and protection of Mexicans was paramount, he was hopeful that ties between the two countries would strengthen.
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Trump speech: 'This is your day'
President Donald Trump uses his inaugural address as the 45th President of the United States to declare the people as the rulers of the nation.
He said he would seek "respectful" dialogue with Mr Trump, who angered many Mexicans on the campaign trail by deriding them as rapists and murderers, and vowing to build a massive wall along the US border to keep them out.
Germany's Vice-Chancellor Sigmar Gabriel warned of a "rough ride" after President Donald Trump was sworn in.
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He said his country would need a new economic strategy geared toward Asia should the new US administration start a trade war with China.
"What we heard today were high nationalistic tones," he said. "I think we have to prepare for a rough ride."
Even so, Mr Gabriel said Europe and Germany must stand together "to defend our interests".
The US is Germany's biggest trading partner and Mr Trump has said he would impose a border tax of 35 per cent on cars that German carmaker BMW plans to build at a new plant in Mexico and export to the US.
There were no such concerns in Moscow, where Champagne flowed and Russians expressed optimism for better relations with Washington.
"It's weird, but it's great, and for the first time ever Russians are applauding the victory of a US presidential candidate. It's a sign of the times," political analyst Stanislav Byshok said.
Mr Trump's promises to fix ravaged relations with Moscow have elated Russia's political elite following spiralling tensions with Washington over the Ukrainian crisis, the war in Syria and allegations of Russian meddling in the US election.
"We are ready to do our share of the work in order to improve the relationship," Prime Minister Dmitry Medvedev said on Facebook.
About 100 Trump sympathisers, nationalists and spin doctors gathered at a hipster party just a few hundred metres away from the Kremlin to celebrate, with a triptych of Mr Trump, Russian President Vladimir Putin and French far-right Front National leader Marine Le Pen in the centre of the hall.
An hour before Mr Trump took to the stage in Washington, the sound of opening Champagne bottles echoed in the hall.
Some party-goers wore Guy Fawkes masks, associated with hackers, in a sly reference to charges that Russia interfered in the US election.
Mr Trump's praise for Mr Putin has raised expectations that he could move to normalise ties, although Mr Trump hasn't articulated a clear policy and some of his cabinet nominees have made hawkish statements on Russia.
Leonid Slutsky, the head of the foreign affairs committee in the lower house of parliament, expressed hope that Mr Trump would establish constructive ties with Moscow.
"We expect a slow but steady revival of our relations," he said.
Speaking at the World Economic Forum in Davos, Switzerland, Russian Deputy Prime Minister Igor Shuvalov voiced hope that Trump will work with Putin on solving the Ukrainian crisis and other problems, but warned against expectations of quick progress. "Difficulties will remain," he said.
'Trump committed to UK relations'
British Prime Minister Theresa May said that even though Mr Trump pledged to put "America first", the new US President was committed to strengthening the "special relationship" between the UK and the United States.
In his inaugural address, Mr Trump promised that "every decision on trade ... will be made to benefit American workers and American families".
But Mrs May said Mr Trump wanted to advance links between Britain and the US.
She hopes to secure a post-Brexit free trade deal with the US to help offset any economic hit caused by leaving the European Union.
Mrs May is expected to visit Mr Trump later this year.
Canada sidesteps trade fears
North of the border, Canada's government, eager to sidestep potentially protectionist US policies, said it was confident Mr Trump would see that working closely with Canada benefited both economies.
Mr Trump – who is committed to renegotiating the North American Free Trade Agreement (NAFTA) with Canada and Mexico – said during his inauguration speech that his administration would stick to a policy of "Buy American."
Canada sends 75 per cent of its exports to the United States and could be crippled by any new US measures designed to restrict imports.
Prime Minister Justin Trudeau issued a statement congratulating Trump. "Together, we benefit from robust trade and investment ties, and integrated economies, that support millions of Canadian and American jobs," he said.
Pope Francis': Let ethics guide you
Pope Francis urged Mr Donald Trump to be guided by ethical values, saying the new president must take care of the poor during his time in office.
"At a time when our human family is beset by grave humanitarian crises demanding farsighted and united political responses, I pray that your decisions will be guided by the rich spiritual and ethical values that have shaped the history of the American people and your nation's commitment to the advancement of human dignity and freedom worldwide," Francis said in a message sent to Trump minutes after he was inaugurated.
"Under your leadership, may America's stature continue to be measured above all by its concern for the poor, the outcast and those in need."
Last year, in response to an answer about Mr Trump's views on immigration and his intention to build a wall on the border with Mexico, Francis said a man with those views was "not Christian".
Mr Trump hit back, calling the pope's comments "disgraceful."
AP, Reuters, AAP