(CNN) Republican candidate Donald Trump's ascension to the White House has sent shock waves around the world. But it appears to have pleased one leader in particular: Russian President Vladimir Putin.

Within hours of Trump's victory speech early Wednesday, Putin congratulated the US president-electand flagged Moscow's willingness to restore ties fully with its old Cold War foe, currently at loggerheads over the Syrian conflict.

Russia featured heavily in a heated campaign, with Trump praising Putin as a firm leader as US officials accused Moscow of meddling by leaking hacked Democratic campaign emails to undermine Hillary Clinton's candidacy.

"(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."

Winter Is Here. — Garry Kasparov (@Kasparov63) November 9, 2016

Diplomatic headaches

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 ."

Photos: Donald Trump's rise President-elect Donald Trump has been in the spotlight for years. From developing real estate and producing and starring in TV shows, he became a celebrity long before winning the White House. Hide Caption 1 of 37 Photos: Donald Trump's rise Trump at age 4. He was born in 1946 to Fred and Mary Trump in New York City. His father was a real estate developer. Hide Caption 2 of 37 Photos: Donald Trump's rise Trump, left, in a family photo. He was the second-youngest of five children. Hide Caption 3 of 37 Photos: Donald Trump's rise Trump, center, stands at attention during his senior year at the New York Military Academy in 1964. Hide Caption 4 of 37 Photos: Donald Trump's rise Trump, center, wears a baseball uniform at the New York Military Academy in 1964. After he graduated from the boarding school, he went to college. He started at Fordham University before transferring and later graduating from the Wharton School, the University of Pennsylvania's business school. Hide Caption 5 of 37 Photos: Donald Trump's rise Trump stands with Alfred Eisenpreis, New York's economic development administrator, in 1976 while they look at a sketch of a new 1,400-room renovation project of the Commodore Hotel. After graduating college in 1968, Trump worked with his father on developments in Queens and Brooklyn before purchasing or building multiple properties in New York and Atlantic City, New Jersey. Those properties included Trump Tower in New York and Trump Plaza and multiple casinos in Atlantic City. Hide Caption 6 of 37 Photos: Donald Trump's rise Trump attends an event to mark the start of construction of the New York Convention Center in 1979. Hide Caption 7 of 37 Photos: Donald Trump's rise Trump wears a hard hat at the Trump Tower construction site in New York in 1980. Hide Caption 8 of 37 Photos: Donald Trump's rise Trump was married to Ivana Zelnicek Trump from 1977 to 1990, when they divorced. They had three children together: Donald Jr., Ivanka and Eric. Hide Caption 9 of 37 Photos: Donald Trump's rise The Trump family, circa 1986. Hide Caption 10 of 37 Photos: Donald Trump's rise Trump uses his personal helicopter to get around New York in 1987. Hide Caption 11 of 37 Photos: Donald Trump's rise Trump stands in the atrium of the Trump Tower. Hide Caption 12 of 37 Photos: Donald Trump's rise Trump attends the opening of his new Atlantic City casino, the Taj Mahal, in 1989. Hide Caption 13 of 37 Photos: Donald Trump's rise Trump signs his second book, "Trump: Surviving at the Top," in 1990. Trump has published at least 16 other books, including "The Art of the Deal" and "The America We Deserve." Hide Caption 14 of 37 Photos: Donald Trump's rise Trump and singer Michael Jackson pose for a photo before traveling to visit Ryan White, a young child with AIDS, in 1990. Hide Caption 15 of 37 Photos: Donald Trump's rise Trump dips his second wife, Marla Maples, after the couple married in a private ceremony in New York in December 1993. The couple divorced in 1999 and had one daughter together, Tiffany. Hide Caption 16 of 37 Photos: Donald Trump's rise Trump putts a golf ball in his New York office in 1998. Hide Caption 17 of 37 Photos: Donald Trump's rise An advertisement for the television show "The Apprentice" hangs at Trump Tower in 2004. The show launched in January of that year. In January 2008, the show returned as "Celebrity Apprentice." Hide Caption 18 of 37 Photos: Donald Trump's rise A 12-inch talking Trump doll is on display at a toy store in New York in September 2004. Hide Caption 19 of 37 Photos: Donald Trump's rise Trump attends a news conference in 2005 that announced the establishment of Trump University. From 2005 until it closed in 2010, Trump University had about 10,000 people sign up for a program that promised success in real estate. Three separate lawsuits -- two class-action suits filed in California and one filed by New York's attorney general -- argued that the program was mired in fraud and deception. Trump's camp rejected the suits' claims as "baseless." And Trump has charged that the New York case against him is politically motivated. Hide Caption 20 of 37 Photos: Donald Trump's rise Trump attends the U.S. Open tennis tournament with his third wife, Melania Knauss-Trump, and their son, Barron, in 2006. Trump and Knauss married in 2005. Hide Caption 21 of 37 Photos: Donald Trump's rise Trump wrestles with "Stone Cold" Steve Austin at WrestleMania in 2007. Trump has close ties with the WWE and its CEO, Vince McMahon. Hide Caption 22 of 37 Photos: Donald Trump's rise For "The Apprentice," Trump was honored with a star on the Hollywood Walk of Fame in January 2007. Hide Caption 23 of 37 Photos: Donald Trump's rise Trump appears on the set of "The Celebrity Apprentice" with two of his children -- Donald Jr. and Ivanka -- in 2009. Hide Caption 24 of 37 Photos: Donald Trump's rise Trump poses with Miss Universe contestants in 2011. Trump had been executive producer of the Miss Universe, Miss USA and Miss Teen USA pageants since 1996. Hide Caption 25 of 37 Photos: Donald Trump's rise In 2012, Trump announces his endorsement of Republican presidential candidate Mitt Romney. Hide Caption 26 of 37 Photos: Donald Trump's rise Trump speaks in Sarasota, Florida, after accepting the Statesman of the Year Award at the Sarasota GOP dinner in August 2012. It was shortly before the Republican National Convention in nearby Tampa. Hide Caption 27 of 37 Photos: Donald Trump's rise Trump appears on stage with singer Nick Jonas and television personality Giuliana Rancic during the 2013 Miss USA pageant. Hide Caption 28 of 37 Photos: Donald Trump's rise In June 2015, during a speech from Trump Tower, Trump announced that he was running for President. He said he would give up "The Apprentice" to run. Hide Caption 29 of 37 Photos: Donald Trump's rise Trump -- flanked by U.S. Sens. Marco Rubio, left, and Ted Cruz -- speaks during a CNN debate in Miami on March 10. Trump dominated the GOP primaries and emerged as the presumptive nominee in May. Hide Caption 30 of 37 Photos: Donald Trump's rise The Trump family poses for a photo in New York in April. Hide Caption 31 of 37 Photos: Donald Trump's rise Trump speaks during a campaign event in Evansville, Indiana, on April 28. After Trump won the Indiana primary, his last two competitors dropped out of the GOP race. Hide Caption 32 of 37 Photos: Donald Trump's rise Trump delivers a speech at the Republican National Convention in July, accepting the party's nomination for President. "I have had a truly great life in business," he said. "But now, my sole and exclusive mission is to go to work for our country -- to go to work for you. It's time to deliver a victory for the American people." Hide Caption 33 of 37 Photos: Donald Trump's rise Trump faces Democratic nominee Hillary Clinton in the first presidential debate, which took place in Hempstead, New York, in September. Hide Caption 34 of 37 Photos: Donald Trump's rise Trump apologizes in a video, posted to his Twitter account in October, for vulgar and sexually aggressive remarks he made a decade ago regarding women. "I said it, I was wrong and I apologize," Trump said, referring to lewd comments he made during a previously unaired taping of "Access Hollywood." Multiple Republican leaders rescinded their endorsements of Trump after the footage was released. Hide Caption 35 of 37 Photos: Donald Trump's rise Trump walks on stage with his family after he was declared the election winner on November 9. "Ours was not a campaign, but rather, an incredible and great movement," he told his supporters in New York. Hide Caption 36 of 37 Photos: Donald Trump's rise Trump is joined by his family as he is sworn in as President on January 20. Hide Caption 37 of 37

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.

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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."

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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.

I hand over the mantle to @RealDonaldTrump! Many congratulations. You have fought a brave campaign. pic.twitter.com/txD3RFMQ2l — Nigel Farage (@Nigel_Farage) November 9, 2016

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.