In their statement on Wednesday, Harry and Meghan said they made the decision "after many months of reflection and internal discussions".
They said they intend "to step back as 'senior' members of the Royal Family, and work to become financially independent".
They plan to split their time between the UK and North America, while "continuing to honour our duty to the Queen, the Commonwealth, and our patronages".
"This geographic balance will enable us to raise our son with an appreciation for the royal tradition into which he was born, while also providing our family with the space to focus on the next chapter," the couple said.
The couple have just returned after a six-week break from royal duties, which they spent in Canada with their eight-month-old son, Archie.
After returning to the UK on Tuesday, Harry, 35, and Meghan, 38, visited Canada's High Commission in London to thank the country for hosting them and said the warmth and hospitality they received was "unbelievable".
Former actress Meghan lived and worked in Toronto for seven years during her time starring in the popular US legal drama Suits.
Under current rules, the pair said they are "prohibited from earning income in any form", but in their new roles could follow other title-holding royals in having full-time jobs.
The costs of official overseas visits will be funded by the Sovereign Grant and contributions from the host country "when appropriate".
The couple are classified as "internationally protected people", which means they must have armed security provided by the Metropolitan Police.
They will retain Frogmore cottage, the Grade-2 listed property in Windsor that cost taxpayers £2.4m to renovate, as their official residence so they have a "place to call home" in the UK.
What do commentators say?
Bryony Gordon, a Daily Telegraph journalist who knows and has interviewed the couple, suggested their decision could be linked to their mental health, after becoming a "punching bag" for a "misogynistic and racist" nation.
Speaking to Emma Barnett on BBC 5 Live, she said: "Here is a man who everyday has to live out the trauma that he experienced as a little boy when he had to walk behind her coffin at the age of 12 in front of the world, and I think if any of us were put in that situation we would find it incredibly triggering."
She added the couple had become a "punching bag for a nation that is still sort of deeply misogynistic and racist" and they were "entitled" to "do what is best" for their family's mental health.
She believes the couple will carry on exactly as before "but just for free", as they're "sick of the scrutiny" that comes with taking public money.
Royal historian Prof Kate Williams said it will be "difficult" for Harry and Meghan to lead "normal" lives, as media interest in them will increase after their "unprecedented" move.
She told BBC Breakfast: "It is going to be difficult if Harry and Meghan are going to live in Canada for a certain period of time and try and get on with business and be normal, charitable CEOs.
"They will require security because I don't see the media interest in them waning, I see it as probably increasing, because what they're doing is so unprecedented for royals."
Royal commentator Penny Junor said the couple's actions were reminiscent of those of Harry's mother, Diana, Princess of Wales. The suggestion that they had not consulted other royals before making their statement was "beyond bizarre".
She said: "It has echoes of Diana when she suddenly announced after her separation [from Charles] that she was stepping back from 50 of her charities without consulting anybody.
"The problem is that they are not working for themselves, they are working for a family firm and to be making announcements of this sort without consulting is beyond bizarre."
Graham Smith, a spokesman for Republic, which campaigns for an elected head of state, said Harry and Meghan's decision "raises questions about the monarchy's future" and will prompt taxpayers to ask how the couple's extra security and overseas lifestyle will be funded.
He added: "To suggest that they're not already financially independent is incredibly crass and belies a sense of self-entitlement and a lack of self-awareness that is common among royals."
He added: "This really is wanting to have your cake and eat it. They have said they will dip in and out of royal duties as it suits them but won't stop taking public money until they find other sources of income."
A staging post to leaving the Royal Family?
This is clearly a major rift between Harry and Meghan on one part, and the rest of the Royal Family on the other.
There are far more questions than answers. What will their new role be? Where will they live, and who will pay for it? What relationship will they have with the rest of the Royal Family?
And there's the institutional question - what does this mean for the Royal Family?