Ahead of her visit, Foreign Secretary Boris Johnson said he hoped leaders of the 27 other nations would match the PM's "generous" proposals to guarantee the rights of European Union nationals with a similar offer to the British expats on the Continent.
Mr Varadkar will hold bilateral meetings with the presidents of the European Commission and European Council ahead of the summit.
"Today we agreed on dates, we agreed on organisation, and we agreed on priorities for the negotiation", he said, outlining a two-step negotiation. But Dalia Grybauskaite also joined in the pop lyrics fun on Twitter by suggesting that even after Britain's "divorce", its European allies could still be counted on as friends.
He pushed at the summit for joint European defense, a joint budget for countries that use the euro and a tougher stance against the US and China on trade.
He said that while the European Union could not replace governments in fighting home-grown militants, it could put pressure on technology firms to act against "content that spreads terrorist material or incites to violence".
Orban says "there are quite a few veterans here who have been laboring for decades".
"His entree is not very encouraging, because he thought yesterday that the best form of friendship is to kick into the eastern European countries. That's not the norm here".
"If the Brits have problems with the movement of people inside Europe, they could maybe be convinced by being reminded that this is only about the free movement of labour [not all people]. It's a project, an ambition". The leaders are due to discuss counter-terrorism, closer European defense cooperation and migration.
His Belgian neighbor Charles Michel was more cautious about even raising the prospect of Britain staying in the Union: "I am NOT a dreamer and I'm not the only one", Michel told reporters. "And I am not the only one".
"I am therefore confident that our Brexit schedule will not be affected, and will be closely following the negotiations in the coming months".
He added that "What we also need is certainty, for our companies in Belgium, in Europe".
"Paradoxically, the tough challenges of the recent months have made us more united than before", Tusk said.
They were briefed by Michel Barnier, who launched the Brexit negotiations for them on Monday, and discussed the move of two European Union agencies from London after Britain quits. Those have started constructively.
May said "that has been an important issue".
May will discuss the matter at the summit's evening session.
Other leaders offered few details on their reservations. "Not for dreams and uncertainty".
"I am an Anglophile", said Mr Rutte.
Soros was also adamant that Prime Minister Theresa May should seek to keep Britain in the EU's single market.
The U.K.'s stance on the rights of an estimated 4.5 million European and British nationals living in each others' countries is expected to fall short of the EU's expectations, officials in Brussels said.
Leaders had agreed with summit chair Donald Tusk not to open discussions with May and she left immediately afterwards, leaving the other 27 to discuss other Brexit issues without her.
"In Europe, I never have illusions because I don't want to lose them", Juncker said.
On Monday, her Brexit minister, David Davis, described the first day of Brexit talks to unravel more than 40 years of union as setting a "solid foundation" for future discussions.
Tusk has made the comments a few hours before a bilateral meeting with British Prime Minister Theresa May.
"A lot less swagger in Davis by the end of the day than at the start", the Guardian's Brexit policy editor, Dan Roberts, wrote on Instagram.