Labour has lost out in Scotland, falling to an embarrassing third place behind the Conservatives.
Scottish voters hope the Tories will be more willing to stand up to Nicola Sturgeon and the SNP, which led the polls but has lost its majority.
ITV News correspondent Martin Geissler rounds up the day's results.
SNP leader Nicola Sturgeon has confirmed she will not seek to form a coalition with another party.
The SNP (Scottish National Party) fell two seats short of an overall majority - winning 63 of 129 seats available - but with such a "large group of MSPs elected", Ms Sturgeon said she saw no need to seek a "formal arrangement".
Instead, she said, when the Scottish Parliament reconvenes in the coming days, she would ask them to formally re-elect her as First Minister, and she will then form a minority SNP government.
"We won a clear and unequivocal mandate, and I secured the personal mandate I sought to implement the bold and ambitious programme for government that I asked the country to vote for," she said.
On the question of independence, she added, the SNP would make their case " with passion, with patience and with respect", saying their aim was to "persuade, not to divide".
We will always respect the opinion of the people - now and in the future - and we simply ask that other parties do likewise.
To those who voted for me yesterday, thank you from the bottom of my heart. You have given me a precious opportunity to change this country for the better and I promise to seize it with both hands.
To those who did not vote for me, I promise I will never stop striving to earn your trust and support.
She said she had a "duty" to "rise above party politics", and to govern in the best interests of all of the country as a whole.
Prime Minister David Cameron has called Scottish First Minister Nicola Sturgeon to congratulate her on the SNP's election success, a Downing Street spokesman said.
The SNP claimed an historic third successive victory in the Parliamentary elections - though fell just shy of a second overall majority.
In return, Ms Sturgeon congratulated Mr Cameron on the gains made by the Conservatives, who won an all-time high of 31 seats - pushing Labour into third place.
The spokesman said the leaders discussed how they could work together on issues such as those facing the steel industry.
The Prime Minister and Ms Sturgeon agreed that the UK and Scottish Governments must continue to work together constructively, most crucially in the short term on the future of the steel industry.
It was noted that the two governments sharing information and experience could be of benefit to the steel industry across the UK, and they agreed to keep in touch on this issue.
Nicola Sturgeon said she was "desperate" for Scotland to become independent - but insisted that the decision over whether to hold a second referendum would lie with voters.
It comes as the SNP secured an historic third term in charge at yesterday's elections, with 63 seats - but fell just short of the 65 which would have given them a majority for the second time.
Speaking after she retained her Glasgow Southside seat, she said:
Everybody knows I desperately want to see Scotland become an independent country, but the decision on that will always lie where it firmly belongs, and that is in the hands of the Scottish people.
Shadow home secretary Andy Burnham has said that Labour has "serious re-building" to do following the party's slide behind the Tories in the Scottish Parliament elections.
Labour has lost support in Scotland.
The job of rebuilding Scotland ... will have to be hard won over the years, rebuilding that bond of trust that we had with the Scottish electorate, obviously that has been broken and there's longterm job of work to be done to rebuild that.
Nicola Sturgeon declared the SNP has "made history" by winning a third term in government at Holyrood - but the party failed to reach a majority.
Prime Minister David Cameron has congratulated Scottish Tory leader Ruth Davidson an securing an "historic result" for the party.
The Conservatives more than doubled their number of seats in the Scottish Parliament elections, up to 31 from the 15 secured in 2011.
It puts the Tories in second place at Holyrood for the first time since devolution.
The SNP has failed to secure a second majority at Holyrood, winning 63 of the 129 seats at the Scottish Parliament.
The Scottish Conservatives had their best ever result, securing 31 MSPs, while Labour suffered its worst result since devolution with 24 MSPs.
Scottish Secretary David Mundell has hailed a "seismic change" in politics politics north of the border after an election night which looks set to put the Conservatives as the official opposition.
The Dumfriesshire, Clydesdale and Tweeddale MP praised Scotland's Tory leader Ruth Davidson for pushing the party into second place ahead of Labour.
He said the results were "good for Scotland, good for the Scottish Parliament".
Ruth has demonstrated throughout this campaign that she is the one politician that can go toe-to-toe with Nicola Sturgeon, that she can stand up to demands for a second referendum.
She can also scrutinise the SNP and make sure they deliver on promises they have made on health, on education, and I think by doing that we will get better government and that will be good for everyone in Scotland. It will be a seismic change in Scottish politics that the Scottish Conservatives are the second party in the Scottish Parliament.
I was a candidate back in those first elections in 1999, it would have been incredible to think the Scottish Conservatives could have finished ahead of Labour and be the official opposition.
It demonstrates that Ruth has transformed our party in Scotland, taken it forward and given us, I hope, a very significant role in the next Scottish Parliament.
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The Scottish National Party (SNP) may not have secured a second overall majority in the Scottish Parliament - but are by far the largest party, with more seats than the Tories, Labour and Lib Dems combined.
The Scottish elections are designed to make it all but impossible for any one party to have overall control - but the SNP defeated the odds at the last election in 2011, securing the first majority since the opening of Holyrood.
But, with 115 results in from 129 seats, it seems unlikely to repeat the feat.
So far, the SNP has 60 seats, while the Conservatives have 25, Labour have 20, the Liberal Democrats have four and Scottish Greens have six.
The SNP need 65 seats for a majority.