The couple have been married since 2005 and have three children between the ages of three and nine.
"We go forward with love and friendship for one another and a commitment to co-parenting our children," they said.
Affleck, 42, plays Batman in the forthcoming Batman v Superman film. Garner, 43, has had roles in Juno and Dallas Buyers Club.
Affleck has enjoyed success as both an actor and director. He won an Oscar for best screenplay for Good Will Hunting in 1997, which he wrote with Matt Damon, and another for best picture in 2013 for Argo, which he starred in and directed.
Garner, herself a successful actress, won a Golden Globe for her role as a CIA agent Sydney Bristow in the TV series Alias.
The couple started dating shortly after meeting on the set of Daredevil in 2003, in which they both played superheroes. They married two years later in the Turks and Caicos Islands, and their 10th wedding anniversary was on Monday.
Sources close to the couple say the divorce is amicable and they will share joint custody of their children.
It is reported Affleck will continue to live on the couple's property in a separate house and that the pair will work through their separation with a mediator.
Garner had a previous, four-year marriage to actor Scott Foley, with whom she appeared in the TV series Felicity. They separated in March 2003 and finalised their divorce a year later.
Affleck's former partners include Gwyneth Paltrow and ex-fiance Jennifer Lopez. The couple became known in the press as "Bennifer".
Last year Garner told InStyle magazine her marriage was in a "mindful" stage.
"You can't expect to be courted all the time, and I don't want to court him right now - I don't have the energy.
"But we're definitely in a very mindful place where we're making an effort to be together, do things at the same time and be loving."
During an acceptance speech at the Oscars in 2013, Affleck referred to their marriage as "work".
While collecting the best picture award for Argo, he said to Garner: "I want to thank you for working on our marriage for 10 Christmases.
"It's good. It is work but it's the best kind of work and there's no one I'd rather work with."
The couple, who said they made the decision to split "after much thought and careful consideration", have asked for their privacy to be respected at this "difficult time".
They added: "This will be our only comment on this private, family matter. Thank you for understanding."

Sum: Hollywood couple Ben Affleck and Jennifer Garner say they have made "the difficult decision to divorce".


The group, which was represented by Ed Miliband's barrister wife Justine Thornton, objects to the proposed development near Croydon.
It said Sutton Council had unlawfully acted in approving the waste burner.
But the High Court rejected all grounds of the challenge.
Ms Thornton had told Mrs Justice Patterson that the council had failed to ensure its planning discretion was not influenced by "contractual considerations" for the burner, which will be at Beddington Farmlands, about two miles west of the centre of Croydon.
She said as well as being the local waste planning authority, the authority was also a member of a consortium "in a contractual relationship with the developer of the waste development under challenge".
Ms Thornton said another judge, Mr Justice Collins, who gave permission for the legal challenge to go ahead, had noted that "the potential conflict of interest in Sutton's two roles is obvious".
But Mrs Justice Patterson ruled that the planning authority's decision was lawful and properly justified.
Mr Butcher, who was born in Croydon, said: "When I heard about the incinerator I couldn't quite believe what I was hearing.
"It seems every week news reports indicate that London's air quality is appalling, so it seems crazy that the authorities have approved an incinerator that affects the residents of my hometown of Croydon."
However, the judge upheld the view of the project developer, Viridor Waste (Thames) Ltd, which was given permission for the incinerator in March, that the complaints were "without substance".
She also refused to give permission for a judicial review.
Sutton Council said it was delighted with the decision and that it would ensure the development adheres to the planning permission.
South London Waste Partnership said: "Beddington landfill site is almost full and is costing more and more money as the government increases landfill tax to encourage local authorities to use more sustainable ways of disposing of waste."
It added: "What's more the majority of the landfill site will be turned into a new green space with wildlife warden so that over 10 years the area will become a haven for nature.
"Altogether, this new approach will save the four council's an estimated Â£200m over 25 years which can be better spent on helping people that need our support at a time when council budgets are being cut."

Sum: A campaign which has the support of former England cricketer Mark Butcher and JLS singer JB Gill has failed in a bid to stop an incinerator from being built in south London.


The 25-year-old forward was arrested after his car was stopped in Liverpool city centre in the early hours of Saturday, police said.
The Brazil international is due to appear at Liverpool Magistrates Court on 31 January.
The court appearance date is scheduled for the same day that Liverpool face Chelsea at Anfield.
Liverpool signed the forward from Hoffenheim on a five-year deal for about Â£29m in June last year.
He played in last week's win against Everton at Goodison Park.
A Liverpool spokesperson said the club would not be commenting on the issue until the legal process had concluded.

Sum: Liverpool footballer Roberto Firmino has been charged with drink-driving after being arrested on Christmas Eve.


The Fed had cited concerns about weakening global growth and recent stock market volatility.
The FTSE 100 fell 1.34%, Frankfurt's Dax plunged 3.06%, and in Paris the Cac 40 dropped 2.56%.
On Wall Street, stocks were sharply lower, with the Dow Jones down nearly 1% in morning trading.
Brenda Kelly, head analyst at London Capital Group, said: "Markets have taken cues from the US, but uncertainty prevails and choppiness is the only certain result. Deflation is a concern. China is a concern, and oil prices look set to take another leg lower."
Some analysts said the Fed could still raise rates this year.
"The Fed's assessment of the global economic conditions has made investors nervous as uncertainty about the timing of a US rate hike continues. We think that a rate hike could still be announced in December," said Robert Parkes, equity strategist at HSBC.
But Michael Hewson, chief strategist at CMC Markets, said: "The Fed's rather downbeat outlook came as an unwelcome surprise, and it's likely to take a while for investors to figure out whether the Fed is seeing something that the rest of us aren't."

Sum:
European and US markets have fallen following the Federal Reserve's decision on Thursday not to raise US interest rates.