Tim Farron has said he is "gutted" and "heartbroken" by the leave result in the EU referendum.
While the Liberal Democrat leader said he will "respect" the outcome of the vote, he added that the "fight to build a better Britain starts here".
Margaret Hodge has told ITV News that Jeremy Corbyn's support for Britain to remain a part of the European Union was "half-hearted" as she called on him to step down.
He has "failed" in his leadership, the Barking & Dagenham MP said.
Speaking after tabling a motion of no confidence in the Labour leader, she said the party's voters had been muddled by Labour's message during the campaign.
They described him as "at best nonplussed, at worst pleased" with the vote for Britain to leave the European Union.
Labour Shadow Minister on Corbyn: we can't go on the way we are. Members are 'appealing to Shadow Cabinet to take action'
Labour Shadow Minister: dozens of emails from party members calling Corbyn to go. 'He's at best non-plussed, at worst pleased' with #Brexit
Senior EU representatives have issued a direct challenge to Britain to implement the terms of its Brexit "as soon as possible, however painful that process may be".
Vote Leave leaders Boris Johnson and Michael Gove earlier said there was "no need for haste" in invoking the Clause 50 process to trigger Britain's formal two-year withdrawal.
But a joint statement, led by European Council president Donald Tusk and European Commission president Jean-Claude Juncker warned "any delay would unnecessarily prolong uncertainty".
We have rules to deal with this in an orderly way. Article 50 of the Treaty on European Union sets out the procedure to be followed if a member state decides to leave the European Union.
We stand ready to launch negotiations swiftly with the United Kingdom regarding the terms and conditions of its withdrawal from the European Union.
A woman who voted to Leave the European Union on Thursday has said that she is "very disappointed" by the results.
The woman, identified only as Mandy, spoke at Manchester Airport, saying that "reality" has now hit her and that given her chance again she would vote to remain.
The student said her decision was due to "the pressure of being told" by the two competing sides who she should vote for.
Two Labour MPs have submitted a motion of no confidence in party leader Jeremy Corbyn.
Margaret Hodge tabled the motion, her office has confirmed. It was seconded by Ann Coffey.
Earlier, Labour Leave campaigner Gisela Stuart used a victory address to call on party leaders to "reflect on how they represent the views of voters."
Without naming Mr Corbyn, she said: "It is now incumbent to all party leaders to also reflect to what extent they are representing and reflecting the views of the voters."
Turkey's prime minister has called on the EU to "reconsider its vision" in the wake of Britain's exit the EU referendum as world leaders and even the Pope responded to the historic vote.
Binali Yildirim said European leaders should read the Brexit vote properly as he continued to press for his nation's admittance into the member bloc.
The issue of Turkey's potential future entry into the EU formed a key dispute between the Vote Remain and Vote Leave campaigners in the build up to Thursday's historic vote.
Pope Francis meanwhile said Britain's EU exit was the "will expressed by the people" but must be followed by "guarantees" for the good of both Britain and countries on the continent.
South African president Jacob Zuma said his country could withstand the financial shocks caused by the Brexit.
Zuma said the treasury and central bank were in talks with financial institutions on the possible implications on Africa's most industrialised country.
Nicola Sturgeon said she and London Mayor Sadiq Khan share the same view about their areas remaining in EU.
I have also spoken this morning with Sadiq Khan and he is clear that he shares this objective for London, so there is clear common cause between us.
German chancellor Angela Merkel has said it must the goal for the European Union to maintain "close future relations" with Britain as she confirmed she will hold a summit of leaders on Monday.
Mrs Merkel said she will meet with the European Council president Donald Tusk, French president Francois Hollande and Italy's prime minister Matteo Renzi in Berlin on Monday.
The German leader said she "deeply regrets" the decision of a majority of British voters at the ballot box to split from the 28-member bloc.
But amid apparent anxiety from other leaders, she said Europe needs to stay "calm and composed" in the wake of Britain's exit.
Cornwall has issued an urgent plea for reassurance that it will not be worse off following the Brexit vote.
The county has received a "significant amounts" of funding from the EU for the past 15 years due to its "relatively weak economy".
But, after 56.5% of voters in the county chose to leave the Union, the council says it is now seeking urgent reassurance that money allocated to it will still be received.
Prior to the vote the Council said they were told by the Leave campaign that funding would still be available.
They also said they had been told Cornwall "would not be worse off" in terms of investment they received.
Now that we know the UK will be leaving the EU we will be taking urgent steps to ensure that the UK Government protects Cornwall’s position in any negotiations.
We will be insisting that Cornwall receives investment equal to that provided by the EU programme which has averaged £60M per year over the last ten years.