The Scottish government has described talks between David Cameron and Scottish First Minister Nicola Sturgeon as "constructive and helpful".
The discussions in Edinburgh were dominated by the issue of further powers for the Scottish Parliament.
The Scottish government said Mr Cameron indicated he was open to considering proposals which would go further than the Smith Commission.
Mr Cameron is expected to comment on the discussions shortly.
A spokesperson for Ms Sturgeon said both sides concluded that the Smith Commission proposals would be implemented "as agreed".
They have also agreed to meet more regularly, possibly quarterly.
The two leaders shook hands on the front steps as Mr Cameron arrived at Ms Sturgeon's official residence at Bute House.
Cameron and Sturgeon talks 'constructive and helpful'
BBC News
15 May 2015
The meeting was held at the Scottish First Minister's official residence at Bute House in Edinburgh
The Scottish government has described talks between David Cameron and Scottish First Minister Nicola Sturgeon as "constructive and helpful".
The discussions in Edinburgh were dominated by the issue of further powers for the Scottish Parliament.
The Scottish government said Mr Cameron indicated he was open to considering proposals which would go further than the Smith Commission.
Mr Cameron is expected to comment on the discussions shortly.
A spokesperson for Ms Sturgeon said both sides concluded that the Smith Commission proposals would be implemented "as agreed".
They have also agreed to meet more regularly, possibly quarterly.
The two leaders shook hands on the front steps as Mr Cameron arrived at Ms Sturgeon's official residence at Bute House.
Bute House in Charlotte Square, Edinburgh, is the official residence of the Scottish First Minister
Mr Cameron left the building shortly before 11:00, almost an hour after he had arrived.
The prime minister, who is visiting Scotland on a post-election tour of the UK, has promised a devolution bill in the Queen's Speech later this month.
But Ms Sturgeon argues that the proposals do not go far enough.
She had been joined for the talks by her deputy John Swinney, who is also the Scottish finance secretary.
Mr Cameron was accompanied by David Mundell, who is the new secretary of state for Scotland, and Andrew Dunlop, who was controversially appointed as a junior minister at the Scotland Office on Thursday.
Mr Cameron and Ms Sturgeon both enjoyed success in last week's general election
Mr Cameron left Bute House about an hour after arriving for the discussions
Ms Sturgeon and Mr Cameron spoke by phone last Friday - the day after election voting - and agreed to hold more detailed discussions at the earliest possible opportunity.
The first minister said she had "made it clear" to Mr Cameron during their brief telephone conversation that "it cannot be business as usual" in Scotland.
Mr Cameron's Conservatives won an overall majority in the election, but Ms Sturgeon's SNP won 56 of the 59 seats in Scotland.
'Crystal clear'
Ahead of their face-to-face meeting, Ms Sturgeon said she was "looking forward to serious and substantial talks" with the prime minister and would have a "constructive and co-operative approach" in dealing with the UK government.
"But, as I have made crystal clear, the general election result last week, and the overwhelming mandate that has given the SNP, means that it simply cannot be 'business as usual' when it comes to Westminster's attitude to Scotland - whether on public spending or on more powers for Scotland," she said.
"The proposals of the Smith Commission are a good starting point, but the election result shows that people all across Scotland are keen to move beyond the extra powers it identified."
The Smith Commission was set up in the wake of September's vote against Scottish independence.
The commission's main recommendation was that Scotland needed powers to set income tax rates and bands on earned income - and to keep all the income tax raised in Scotland.
But Ms Sturgeon also wants powers over employment policy including the minimum wage, welfare, business taxes, National Insurance and equality policy to be devolved as a matter of priority.
The first minister has dismissed newspaper reports which quote a "senior SNP source at Westminster" as saying the party could push ahead with a second independence referendum without the consent of Westminster.
Her representative said: "These claims are totally wrong - there are no such plans. The position is crystal clear: the general election was not a mandate for another referendum. And there will only be another referendum if and when the people of Scotland back such a proposal at a Scottish Parliament election."
Fiscal autonomy
Some Conservative politicians have urged Mr Cameron to consider offering a more radical package of devolution to Ms Sturgeon, including full fiscal autonomy - full control over tax and spending.
But the prime minister has so far given no indications that he intends to go beyond the proposals contained in the Smith agreement.
Mr Cameron said he was visiting Scotland to "underline my commitment to our United Kingdom and Scotland's important place within it".
"That means remaining true to the promise we made to implement the all-party Smith agreement to make Scotland one of the most accountable and powerful devolved parliaments in the world," he said.
He added: "Scotland has two governments and it is the duty of the first minister and myself to respect each other's roles and responsibilities and to work together for the benefit of all the people of Scotland.
"As more powers are devolved to Scotland, it is time to move beyond the debate about processes and focus on those bread and butter issues that affect every family in our United Kingdom - jobs, homes, good schools and strong public services, and dignity and respect in retirement."
Analysis by Laura Bicker, Scotland correspondent
What Nicola Sturgeon is trying to say is she has a mandate. She has 56 out of the 59 MPs elected in Scotland and she won 50% of the vote - unprecedented here in Scotland.
She says that gives her leverage to push David Cameron. David Cameron says he wants to keep the nation together.
The language coming out of Westminster last week was very much a cordial one. 'We will listen'.
Nicola Sturgeon has warned him that if he doesn't listen, then the people of Scotland will judge him accordingly.
Analysis by Brian Taylor, BBC Scotland political editor
It's a question of calculation for Mr Cameron, as it is for Nicola Sturgeon. How far can they make concessions and what level of deal can be reached? The starting point for the prime minister is very much deliver what was in Smith - that's what's been agreed.
David Cameron is saying that any changes to Smith have to come about through parliamentary process. That's tabling amendments in the House of Commons. Given that he has an overall majority, that keeps the power very firmly in his hands.
Nicola Sturgeon takes a different approach. She says there should be the involvement of Civic Scotland and discussion and debate in Scotland.
It is challenging for David Cameron but it is also challenging for Nicola Sturgeon - what is the exit strategy if the prime minister says 'no'?
Does she threaten a second referendum? She has said that she is reluctant to do so and it is not currently in her plans.
Cameron and Sturgeon talks 'constructive and helpful' - BBC News
The discussions in Edinburgh were dominated by the issue of further powers for the Scottish Parliament.
The Scottish government said Mr Cameron indicated he was open to considering proposals which would go further than the Smith Commission.
Mr Cameron is expected to comment on the discussions shortly.
A spokesperson for Ms Sturgeon said both sides concluded that the Smith Commission proposals would be implemented "as agreed".
They have also agreed to meet more regularly, possibly quarterly.
The two leaders shook hands on the front steps as Mr Cameron arrived at Ms Sturgeon's official residence at Bute House.
Cameron and Sturgeon talks 'constructive and helpful'
BBC News
15 May 2015
The meeting was held at the Scottish First Minister's official residence at Bute House in Edinburgh
The Scottish government has described talks between David Cameron and Scottish First Minister Nicola Sturgeon as "constructive and helpful".
The discussions in Edinburgh were dominated by the issue of further powers for the Scottish Parliament.
The Scottish government said Mr Cameron indicated he was open to considering proposals which would go further than the Smith Commission.
Mr Cameron is expected to comment on the discussions shortly.
A spokesperson for Ms Sturgeon said both sides concluded that the Smith Commission proposals would be implemented "as agreed".
They have also agreed to meet more regularly, possibly quarterly.
The two leaders shook hands on the front steps as Mr Cameron arrived at Ms Sturgeon's official residence at Bute House.
Bute House in Charlotte Square, Edinburgh, is the official residence of the Scottish First Minister
Mr Cameron left the building shortly before 11:00, almost an hour after he had arrived.
The prime minister, who is visiting Scotland on a post-election tour of the UK, has promised a devolution bill in the Queen's Speech later this month.
But Ms Sturgeon argues that the proposals do not go far enough.
She had been joined for the talks by her deputy John Swinney, who is also the Scottish finance secretary.
Mr Cameron was accompanied by David Mundell, who is the new secretary of state for Scotland, and Andrew Dunlop, who was controversially appointed as a junior minister at the Scotland Office on Thursday.
Mr Cameron and Ms Sturgeon both enjoyed success in last week's general election
Mr Cameron left Bute House about an hour after arriving for the discussions
Ms Sturgeon and Mr Cameron spoke by phone last Friday - the day after election voting - and agreed to hold more detailed discussions at the earliest possible opportunity.
The first minister said she had "made it clear" to Mr Cameron during their brief telephone conversation that "it cannot be business as usual" in Scotland.
Mr Cameron's Conservatives won an overall majority in the election, but Ms Sturgeon's SNP won 56 of the 59 seats in Scotland.
'Crystal clear'
Ahead of their face-to-face meeting, Ms Sturgeon said she was "looking forward to serious and substantial talks" with the prime minister and would have a "constructive and co-operative approach" in dealing with the UK government.
"But, as I have made crystal clear, the general election result last week, and the overwhelming mandate that has given the SNP, means that it simply cannot be 'business as usual' when it comes to Westminster's attitude to Scotland - whether on public spending or on more powers for Scotland," she said.
"The proposals of the Smith Commission are a good starting point, but the election result shows that people all across Scotland are keen to move beyond the extra powers it identified."
The Smith Commission was set up in the wake of September's vote against Scottish independence.
The commission's main recommendation was that Scotland needed powers to set income tax rates and bands on earned income - and to keep all the income tax raised in Scotland.
But Ms Sturgeon also wants powers over employment policy including the minimum wage, welfare, business taxes, National Insurance and equality policy to be devolved as a matter of priority.
The first minister has dismissed newspaper reports which quote a "senior SNP source at Westminster" as saying the party could push ahead with a second independence referendum without the consent of Westminster.
Her representative said: "These claims are totally wrong - there are no such plans. The position is crystal clear: the general election was not a mandate for another referendum. And there will only be another referendum if and when the people of Scotland back such a proposal at a Scottish Parliament election."
Fiscal autonomy
Some Conservative politicians have urged Mr Cameron to consider offering a more radical package of devolution to Ms Sturgeon, including full fiscal autonomy - full control over tax and spending.
But the prime minister has so far given no indications that he intends to go beyond the proposals contained in the Smith agreement.
Mr Cameron said he was visiting Scotland to "underline my commitment to our United Kingdom and Scotland's important place within it".
"That means remaining true to the promise we made to implement the all-party Smith agreement to make Scotland one of the most accountable and powerful devolved parliaments in the world," he said.
He added: "Scotland has two governments and it is the duty of the first minister and myself to respect each other's roles and responsibilities and to work together for the benefit of all the people of Scotland.
"As more powers are devolved to Scotland, it is time to move beyond the debate about processes and focus on those bread and butter issues that affect every family in our United Kingdom - jobs, homes, good schools and strong public services, and dignity and respect in retirement."
Analysis by Laura Bicker, Scotland correspondent
What Nicola Sturgeon is trying to say is she has a mandate. She has 56 out of the 59 MPs elected in Scotland and she won 50% of the vote - unprecedented here in Scotland.
She says that gives her leverage to push David Cameron. David Cameron says he wants to keep the nation together.
The language coming out of Westminster last week was very much a cordial one. 'We will listen'.
Nicola Sturgeon has warned him that if he doesn't listen, then the people of Scotland will judge him accordingly.
Analysis by Brian Taylor, BBC Scotland political editor
It's a question of calculation for Mr Cameron, as it is for Nicola Sturgeon. How far can they make concessions and what level of deal can be reached? The starting point for the prime minister is very much deliver what was in Smith - that's what's been agreed.
David Cameron is saying that any changes to Smith have to come about through parliamentary process. That's tabling amendments in the House of Commons. Given that he has an overall majority, that keeps the power very firmly in his hands.
Nicola Sturgeon takes a different approach. She says there should be the involvement of Civic Scotland and discussion and debate in Scotland.
It is challenging for David Cameron but it is also challenging for Nicola Sturgeon - what is the exit strategy if the prime minister says 'no'?
Does she threaten a second referendum? She has said that she is reluctant to do so and it is not currently in her plans.
Cameron and Sturgeon talks 'constructive and helpful' - BBC News
