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.
The Tories' leader in Scotland has hailed an "incredible result" which looks likely to put the Conservatives in second place north of the border, ahead of Labour.
Ruth Davidson - who won the Edinburgh Central seat from the Scottish National Party (SNP) - said her campaign had focused on providing a "strong opposition" to the ruling SNP.
She also admitted she realised those who had voted for the party were unlikely to be "true blue Conservatives".
I fully recognise and understand there are many people that have given us their vote for the very first time, not because they're true blue Conservatives, but because there's a job of work they want us to do.
I hope the message that was resonating was of being a strong opposition, to hold the SNP to account, to saying no to a second independence referendum, to respect the decision that our country made and to really focus on the things we're paying a government to focus on, on schools, on hospitals, on public services.
That's what people want.
Labour's leader in Scotland has said she was "heartbroken" by overnight election results which look set to push the party into third place, behind the Tories.
Kezia Dugdale said it was clear even the Conservative candidates had not expected to perform as well as they had.
[I'm] heartbroken, without question. You can see from the reaction of the Tories that they are indeed surprised about how well they've done tonight.
There's no doubt that our defeat for the Labour Party is painful, but it is not the end of our campaign. We will continue to argue for Labour values, Labour ideas and Labour principles.
She also ruled out stepping down, saying "burning through leaders" was "not the answer".
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Scotland's First Minister Nicola Sturgeon regained her seat as MSP for Glasgow Southside on Friday morning.
The Scottish National Party politician polled a huge majority in the seat she has held since 2007.
Scottish Labour leader Kezia Dugdale has failed in her bid to win the Edinburgh Eastern constituency.
Ms Dugdale lost out to the Scottish National Party.