European Union leaders say they will remain united and strive to protect the bloc's interest following Britain's decision to leave.
In a statement, the leaders said: "The Union will act as one and preserve its interests.
"Our first priority will be to minimise the uncertainty caused by the decision of the United Kingdom for our citizens, businesses and member states."
They said they would "start by focusing on all key arrangements for an orderly withdrawal".
The leaders will meet in Brussels in one month, on April 29.
But a senior European diplomat said that the European Union will not seek to punish Britain for leaving.
There has been speculation in the British press since last year's referendum on Brexit that the other 27 EU members could try to extract maximum suffering from the UK in order to discourage others from leaving.
The diplomat dismissed such views, noting that Britain will have to grapple with the fallout from its departure from the EU's single market.
"Leaving the common market will hurt a lot all on its own," he said.
Meanwhile, French centrist presidential candidate Emmanuel Macron says his priority after Britain's formal request to leave the EU would be to protect the bloc and European interests.
Speaking after meeting with London Mayor Sadiq Khan, Mr Macron said that he also believes Britain and the EU will need to remain close, notably in defence, even if they have to rebuild their relationship.
He also noted that Mr Khan is mayor to 200,000 French citizens living in London and said he would work to determine how Brexit will affect them.
Mr Macron said in English that in "some of them will decide to come back, I will be very happy to host them again, but, obviously, we will work together with the U.K. and we will work together with London because they live there".
Estonia's prime minister added that "we cannot allow ourselves to be paralysed by Brexit" and "we must decisively move on together".
Juri Ratas said that "we accept this notice with regret in our hearts", adding the priority "is to reduce the insecurity of the people and companies that could be influenced by the United Kingdom's withdrawal".
In a separate statement, the Estonian government said that triggering Article 50 ended "uncertainty on the UK's intentions on leaving the EU and the exit's time-frame."
And the foreign minister of Luxembourg said a "fair solution" following Brexit is in the interest of both the EU and Britain.
Jean Asselborn said in Serbia's capital that "we cannot punish a country that wants to leave the European Union".
But, he added that "when (British Prime Minister) Theresa May says no deal is better than a bad deal I think that's for both sides, not only for one side".
Mr Asselborn explained that "with Brexit the European Union will lose a little bit of money, that's clear, because the contribution of the UK was substantial".
He also warned that "if you are a member of the European Union, you are member of the European Union until the last moment, and you have to fulfil your engagements."
Sweden's prime minister says that he wants "to see organised and result-oriented negotiations" with Britain, saying good relations with London were "important for Britain, for Europe and for Sweden".
Stefan Lofven said Britain had been "a close and valuable partner in the European Union".
Latvian Foreign Minister Edgar Rinkevics tweeted: "As UK has formally triggered Article 50 we should negotiate in a constructive way to forge a fair deal for both EU and UK."
Sweden's ambassador to Britain tweeted that "Swedes in the UK with concerns and questions related to Brexit are welcome to contact Swedish Embassy".
Chancellor Angela Merkel said Germany will push for the impact of Brexit on Germans and other European Union citizens living in Britain to be as minimal as possible.
She said she wants Britain and the EU to remain "close partners".
Ms Merkel said during a speech in Berlin that the divorce talks must first focus on undoing in an orderly fashion four decades of ties between Britain and the EU.
She said: "Only when these questions are cleared up can we subsequently - but hopefully soon - talk about our future relationship."
Ms Merkel also said the remaining EU member nations will negotiate with Britain "in a fair and constructive manner".
She said: "I hope that the British government will also approach the talks in this spirit."
She added that Theresa May had assured her it would in a phone conversation on Tuesday.