Alyn Smith, the SNP Member of the European Parliament for Scotland, has pleaded with members to "not let Scotland down" after the country voted to Remain in the EU, in contrast to the UK overall.
Speaking in the European Parliament, he said: "There's a lot of things to be negotiated. We will need cool heads and warm hearts.
"But please remember this: Scotland did not let you down. Please, I beg you, do not let Scotland down now."
Sinn Fein MEP Martina Anderson has said if English votes "drag us out of the EU that would be like Britannia waves the rules", as she raised concerns over a border between Northern and the Republic of Ireland.
Speaking in the European Parliament, she said: "We accept, we respect and we will defend the wishes of the people of the north of Ireland.
"However if English votes drag us out of the EU, that would be like Britannia waives the rules.
"There was a democratic vote, we voted to remain. I tell you, that the last thing the people of Ireland need is an EU border with 27 member states stopped right in the middle of it."
French far-right leader Marine le Pen has hailed the UK's decision to leave the European Union.
Speaking at an emergency session of the European Parliament to discuss the fallout from the Brexit vote, the National Front leader told MEPs, "put away those sulky faces, put away those angry looks and rejoice in the free and great emancipation" of the British people.
She said she is "committed to pursuing the path of liberty" for France.
Nigel Farage has told the European Parliament the UK voted to leave the EU because a political union was imposed on them.
"The main reason the United Kingdom voted the way it did is because you have by stealth, deception, without ever telling the truth to the British or the rest of the people's of Europe, you have imposed upon them a political union," he said.
"And when the people in 2005 in the Netherlands and France voted against that political union, rejected the constitution, you simply ignored them and brought the Lisbon Treaty in through the back door."
Nigel Farage was greeted with jeers as he stood in the European Parliament - and taunted MEPs by saying: "You're not laughing at me now".
"When i came here 17 years ago and I said I wanted to lead a campaign to get Britain to leave the European Union, you all laughed at me," he said.
"Well I have to say you're not laughing now, are you?
"And the reason you're so upset, the reason you're so angry has been perfectly clear from all the angry exchanges this morning - you as a political project are in denial."
Nigel Farage and Jean-Claude Juncker embraced in the European Parliament on Tuesday, ahead of the first sitting since last week's EU referendum.
Later the European Commission President asked Mr Farage: "Why are you here?"
Jean-Claude Juncker has said the European Parliament "must respect British democracy" following last week's EU referendum.
Speaking at an emergency session to discuss the Brexit vote, the European Commission President's comment was met with applause by the parliament - and a slow clap by Nigel Farage.
Juncker turned to the UKIP MEP and responded in an amused tone: "That's the last time you are applauding here."
"And to some extent I am surprised you are here," he added. "You were fighting for the exit. The British people voted in favour of the exit, why are you here?"
The outgoing UK Commissioner Lord Hill has received a standing ovation from MEPs in the European Parliament.
He announced on Saturday that he is standing down from his post after the referendum vote for Britain to leave the European Union.
The Dutch politician Jeanine Hennis-Plasschaert has told the European Parliament the outcome of the UK's EU referendum "marks a watershed moment in Europe's history".
However she said "history and geography cannot be changed", adding that the the UK "is and will always be a European nation."
"We share the same values, we harbour the same hopes and will continue to work together, as partners and allies," she said.
"In other words it will be in the interest of us all to ensure a future relationship will be constructive and mutually beneficial.
"No one will benefit from a prolonged period of political limbo. The ball is in the British court."