Big Tory guns return to pressing in-trays and unfinished jobs

George Osborne

George Osborne returns to the Treasury to continue trying to complete the job of balancing Britain's budget, which he initially promised to have done by 2014. Despite his difficulties midway through the last term, when he was criticised for stumbling growth levels and enacting the so-called "omnishambles Budget", he returns with his reputation enhanced.

The chancellor's first job will be to enact even steeper spending cuts than he imposed during his first five years. But he may leave much of that work to whomever replaces Danny Alexander as his chief secretary, for he will also take on a central role in renegotiating the terms of Britain's membership of the EU. Completing that task and selling the new arrangement to Conservative MPs is likely to take up much of his next two years.

Mr Osborne also becomes de facto deputy prime minister, having taken on William Hague's former title of first secretary of state.

Theresa May

Like Mr Osborne, Theresa May returns to her job still trying to achieve what she said was her first priority in 2010 - in her case, bringing net migration down below 100,000. Many were surprised that the Conservatives kept this promise in their manifesto, given how far they fell short of this target during their first five years.

She will also return to some of the legislative agenda that was blocked by the Liberal Democrats in coalition government, especially the "snoopers' charter". Ms May wants mobile phone and internet companies to keep records of customers' messages and internet usage for a year. But she may find this more difficult than expected given that some Tories are opposed.

Ms May is also likely to try and give authorities more powers to deal with groups they believe are extremists, such as the ability to close down certain mosques altogether.

Philip Hammond

Philip Hammond spent the first few months in his job as foreign secretary touring the capitals of Europe in an attempt to shore up support for a British renegotiation package. That attempt was reasonably successful - most European leaders say they can live with what Mr Cameron is proposing, although they are very unlikely to bow to the UK demand for treaty change. Mr Hammond's role in Europe will be slightly undermined by the fact that Mr Osborne is seen as the more senior member of the negotiating team, however.

The foreign secretary's in-tray has plenty of other pressing priorities, not least how to deal with the threat of Islamist extremist fighters in Syria and northern Iraq. And he, alongside Ms May, will have to try and find a solution to the problem of refugees fleeing Africa for Europe via Libya; more than 1,000 of them have died in the last year trying to reach the EU.

Mr Hammond will also be keen to keep pressure on European leaders for a tough range of sanctions against Russia as long as it is still interfering in Ukraine.

Michael Fallon

Michael Fallon played an important role during the election campaign, not least with his highly personal attack on Ed Miliband over Labour's approach to renewing the Trident nuclear weapons system.

While the Tory majority means they should face few problems proceeding with the "maingate" decision - the key approval - to renew Trident, which is due next year, he will have more difficulties over army reform and overall levels of defence spending.

On the army, Mr Fallon must boost the recruitment of reservists, which has lagged behind in recent months despite a huge drive to increase numbers while the size of the full-time army is cut.

On defence spending, Mr Fallon faces one of two very difficult options. Either he must persuade Mr Osborne to allocate enough money to the military to make sure Britain continues to spend 2 per cent of its national output on defence, or he must explain to his own party why the government is not doing so.

Michael Gove

Michael Gove brings his zeal for reform of the public sector to the prisons system, a task made harder by Conservative resistance to doing anything that looks anything less than tough on law and order. When Ken Clarke tried to bring in more community sentencing as a way of easing the burden on prisons, he was vigorously opposed by some in his own party.

But Mr Gove's biggest difficulty might be repealing the Human Rights Act, with opponents almost certain to raise a legal challenge against such a move. They argue that a repeal would go against EU law but the Conservatives say they are willing to do it anyway even if it means facing a judicial review.

In his first few days, Mr Gove will also have to try and placate lawyers thinking of going on strike over further cuts to legal aid.

Chris Grayling

Chris Grayling takes over from William Hague as Leader of the House of Commons. While this job does not often hit the headlines, there is scope for slip-ups, as Mr Hague showed when he tried to change the voting system for the Commons Speaker only to be defeated by a major Tory rebellion.

Mr Grayling's first job will be to push through proposals for "English votes for English laws", under which the Tories intend to introduce a new voting stage in the Commons for English and Welsh MPs only. This will have to be done in time for next year's Budget, with Mr Cameron having promised an "English income tax" on which Scottish MPs would be denied a vote at this new stage.

Mr Grayling will also soon begin legislating for a referendum to be held on Britain's membership of the EU. While this will be supported by his party, there may be some fighting over who should vote and when the ballot should take place.

Mark Harper

The job of Conservative chief whip will be made easier in this parliament than it was in the last because by governing alone, Mr Cameron now has more ministerial jobs to offer as incentives to toe the party line. Mark Harper is also well liked in the party, especially after he resigned from his job as immigration minister as soon as it became clear he was likely to face a storm over employing someone without permission to work in the UK.

Mr Harper will have to pass votes with a majority of just 12, however, something that is likely to be especially difficult if the party starts to split over Europe.

Other high-profile names have suffered in the chief whip role before, including Mr Gove, who lost votes on more than one occasion having misread the mood in his own party. The last parliament was one of the most rebellious ever and Mr Harper is likely to face similar problems with some of his own backbenchers.

Nicky Morgan

Nicky Morgan continues in her role as education secretary, where she has been tasked with continuing with Michael Gove's education changes while placating some of the teachers who reacted so angrily to them.

Ms Morgan is likely to oversee a continuing expansion of academies and free schools, while also making further changes to the curriculum, having advocated tougher exams and a focus on broader character development.

But she is not likely to go as far as some believe Mr Gove had wanted in forcing schools to convert to academies, or in allowing companies to run profitmaking schools.

At some point Ms Morgan will have to make a decision on whether the party should allow grammar schools, despite evidence showing they hamper social mobility. And she will also have to tackle the problems in the schools' career advice system, which has suffered over the past five years through a lack of funding.

Iain Duncan Smith

Iain Duncan Smith will continue trying to roll out his ambitious plan to incorporate the six main benefits payments into a single system, known as the Universal Credit.

The work and pensions secretary initially planned to have the scheme working for new claimants nationwide by 2013, but after a series of problems the scheme is not now due to be rolled out across the country until next year.

The other major task on his plate is to help find the £12bn of welfare cuts that the party has promised to deliver in the first two years of government. That was made more difficult during the campaign when the prime minister ruled out cutting child benefit in any way.

Mr Duncan Smith was nearly moved in last year's reshuffle but resisted any attempt to move him into a new role. His retention in post after the election represents a personal victory.

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