"(Trump) spoke about resuming and restoring relations with Russia. We understand the way to that will be difficult, taking into account the current state of degradation of relations between the US and Russia," Putin told Russian state television.
"As I have repeatedly said, that is not our fault that Russia-US relations are in that state. Russia is ready and wants to restore the fully fledged relations with the US. I repeat we understand this will be difficult, but we are ready to play our part in it."
But not all Russians were convinced. Garry Kasparov, the Russian chess grandmaster and fierce Putin critic, tweeted his reaction to Trump's victory: "Winter Is Here." Kasparov wrote a critical book about Putin called "Winter Is Coming."
The uncertainty about what a Trump presidency will mean for diplomatic relations was reflected in anxious statements from other parts of the world, particularly Europe.
European Council President Donald Tusk said: "While respecting the democratic choice of the American people, we are at the same time aware of the new challenges that these results bring. One of them is this moment of uncertainty over the future of our transatlantic relations."
"The events of the last months and days should be treated as a warning sign for all who believe in liberal democracy," he said. "This means that we should finally get our act together and bring back a sense of direction, bring back confidence, bring back a sense of order. Also in the global context whether we talk about trade, migration or security."
French President François Hollande's reaction was also far from enthusiastic.
"The people of America have spoken. I have congratulated Mr. Trump, as it is usual in this situation. I thought of Clinton, with whom I worked during the Obama administration. This result leads to uncertainty," he said, adding that the United States remained an ally of France's.
"I also urge vigilance because of statements made by Donald Trump," he said.
Meanwhile, Cuban President Raul Castro sent a congratulatory message to Trump. The President-elect also heard from Turkish President Recep Tayyip Erdogan, with whom he spoke by phone.
Trump may have to smooth out his relationship with China after he painted the Asian powerhouse as a threat to American jobs during his campaign, making statements such as the United States
"can't continue to allow China to rape our country
Nonetheless, Chinese President Xi Jinping congratulated him in a phone call and later told state-run China Central Television that the two countries "shoulder a special responsibility" as the world's largest developed and developing nations.
"China pays high attention to the Sino-US relationship and hopes to develop a sound, long-term and stable relationship with the US," Xi said.
He added he hoped to "settle all disputes with the US in accordance with the principle of nonconfrontation."
The United States under the Obama administration has built up its presence in the Asia-Pacific region. It regularly carries out military exercises with countries in dispute with Beijing over territory and resources in the South China Sea.
Japanese Prime Minister Shinzo Abe had this to say about the election: "The stability of the Asia-Pacific region, which is a driving force of the global economy, brings peace and prosperity to the United States. Japan and the United States are unwavering allies tied firmly with the bond of universal values such as freedom, democracy, basic human rights and the rule of law."
Issue of US-Mexico wall
Trump sparked outrage from Mexico during his presidential bid, describing Mexican immigrants as rapists and drug dealers, and proposing a wall between the two countries at Mexico's expense to stem the flow of migrants.
Former Mexican President Vicente Fox told CNN he was "shocked by the news" and said his country would never pay for Trump's proposed wall on the US-Mexico border.
"Of course Mexico will never pay for that f***ing wall," Fox told CNN's John Defterios at the Global Business Forum on Latin America
in Dubai, United Arab Emirates.
"He's been very ignorant about the difference between being a mediocre businessman and running a nation, especially a nation like the United States."
But the current Mexican President, Enrique Peña Nieto, struck a more conciliatory tone.
"Mexico and the US are friends, partners and allies and we should continue to collaborate for the competitive development of North America," he wrote on Twitter.
Peña Nieto said he has agreed to a meeting with Trump, possibly during the transitional period.
The Mexican leader said: "Mexico and the US are allies, associates, and neighbors. When Mexico does well, the US does well. And when the US does well, Mexico does well. With this in mind, we've agreed that our teams will be in touch to start designing a new work agenda, one that incorporates themes of common interest such as security, cooperation, and our societies' prosperity."
Venezuela President Nicolas Maduro and US Secretary of State John Kerry spoke on the phone in a scheduled call on Wednesday, and Maduro asked Kerry to pass his congratulations to Trump, Venezuela's foreign ministry announced. Maduro underlined the need to "establish a positive work agenda for the next administration."
Trump has also proposed a blanket ban on Muslims entering the country.
British Prime Minister Theresa May was diplomatic in her congratulations, saying the two nations shared a "special relationship based on the values of freedom, democracy and enterprise."
"We are, and will remain, strong and close partners on trade, security and defense," she said.
Far-right parties celebrate
Europe's far-right party leaders are cheering Trump's win, including Britain's Nigel Farage, the outgoing leader of the UK Independence Party, and France's Marine Le Pen, who sent Trump a congratulatory tweet early Wednesday, adding a pat on the back for the "free" American people.
Le Pen's father, Jean-Marie Le Pen, the founder of the National Front, took to Twitter to say, "Long live President Trump!" -- and claim Trump as part of a worldwide populist wave.
"The American people want Donald Trump to be the people's president. Today the United States, tomorrow France. Bravo!" Le Pen wrote.
Far-right leaders in the Netherlands, Belgium, Russia, the Czech Republic, Italy and Serbia, among other places, also have voiced support for Trump. The hard-right Greek party, Golden Dawn, went so far as to make a pro-Trump video with neo-Nazis.