How you decide what happens to your organs after death depends on where you live in the UK. So what is the current system and what might change?
The change has for now been rejected in England and Northern Ireland, which instead relies on public awareness campaigns.
Three people in the UK die a day in need of donors, according to NHS Blood and Transplant. This is despite an all-time high in registered donors, with 23.4 million people - or 36% of Britons - on the organ register.
Opting-out: What happens in Scotland and Wales
The "soft" opt-out system planned for Scotland means parts of an adult's body can be used in transplants - unless they have opted out. Families and next of kin can still veto the removal of organs.
So-called "deemed consent" has been the system in Wales since December 2015, where 6% of the population has chosen to opt out.
One year on in Wales, 39 organs were transplanted through deemed consent out of a total of 160 transplants.
But deemed consent does not apply to certain people including children and those who lack the capacity to understand the system.
The campaign for changing the law in the UK
Supporters said Scotland's move was a "landmark moment" and the UK government has hinted at the possibility of a future change in England.
Currently in England and Northern Ireland, people must join the NHS Organ Donor Register or their family or close friends must give consent before their organs can be used.
No 10 said it was "keeping a close eye on the implications of the policy change in Wales and Scotland", in reaction to a newspaper campaign to alter the rules across the UK.
"Max Johnson is nine and needs a new heart," the Daily Mirror said on its front page on Friday, adding that opt-out would "give him and thousands of others a far, far greater chance of life".
Although organ donation did not feature prominently in this year's general election, politicians have previously voiced their support for opt-out.
Labour's deputy leader Tom Watson has argued that many people want to donate - but forget to sign up.
But some Conservatives, including the Welsh Conservatives' shadow health minister Darren Millar, have criticised the opt-out policy, saying it was not a "silver bullet" solution.
Church leaders have also expressed concerns about opt-out, saying that donating organs should be seen as a free gift. But most major faith groups - Christians and Muslims included - have expressed support for the principles of donation.
Why isn't it happening in England and Northern Ireland?
Deemed consent has so far been rejected as a UK-wide policy.
In 2008, a task force set up by the Labour government rejected opt-out for the time being - saying it could put donors off or erode people's trust in doctors.
While it did not rule out opt-out, it suggested a £4.5m public awareness campaign instead - adding that families get an emotional benefit from "choosing" to donate their deceased relative's organs.
In Northern Ireland, senior clinicians have argued that educating the public to sign up to donate was the best way forward.
With 23.4 million Britons registered, registered donor numbers have increased every year since 2012 - and are 20% higher than five years ago.
What's it like to wait?
Kevin Mashford, 40, waited a year to receive a life-saving heart transplant and thinks opt-out is the fairer system.
"If you're willing to receive someone's organs to save your life, why not be willing to donate?" he said.
Mr Mashford, an architect from Bristol, was born with a rare heart disease but in 2013, aged 36, he had final stage heart failure and was told he had six months to live.
"The doctors told me to see my family and sort out my affairs," he said. "I waited at home for that year, waiting for news".
He said he was saved in the "nick of time" after going to hospital with multiple organ failure just six weeks before he found a donor.
"I was lucky in a way, they put me on an urgent Europe-wide list," he said. There were 6,402 people on waiting lists for organ transplants in December 2016, according to the NHS.
Whether the system is opt-in or opt-out, Mr Mashford urged people to tell their family their wishes. "Hospitals also have a moral obligation, and if a mum doesn't know if their son is registered and objects to donation, it can be difficult to question that," he said.
Mr Mashford later found out his heart donor was a keen cyclist and began riding in his memory.
"After I recovered, I cycled seven minutes for seven days and I haven't looked back. Since then, with my donors name inscribed in my helmet, I have taken part in a number of long distance cycle rides for charity."
Have we become more likely to donate?
The rise in donations follows efforts by celebrities and public health officials to encourage people to donate, and donation has become more prominent in popular culture.
The NHS runs social media campaigns and makes posters encouraging people to donate.
In 2015, it recruited Made In Chelsea's Jamie Laing and Olympic medallist Jade Jones to make their own dating profiles on the app Tinder, where they sent a message to users asking them to donate.
Others may remember the BBC's That's Life! documentary in the early 1980s, which featured an appeal by the parents of two-year-old Ben Hardwick.
Their wait for a liver transplant earned him celebrity status. He got the transplant but later died in hospital.
In 1999, the award-winning children's book Pig Heart Boy by Malorie Blackman, about a boy with heart failure, was adapted into a children's television series.
As Scotland's Public Health Minister Aileen Campbell said this week, there has been a "long-term culture change in attitudes" among Britons towards donating organs.