Jones-Buchanan, 37 has signed a one-year deal with 32-year-old Ablett committing himself to the club for the next two seasons.
Briscoe is now contracted until the end of 2020.
As previously reported in the Yorkshire Evening Post, Jones-Buchanan and Briscoe attended the first day of pre-season this week.
Ablett is due to begin preparations for the new campaign on Monday.
Jones-Buchanan and Ablett were out of contract at the end of the 2018 season and Briscoe joined Rhinos from Featherstone Rovers - for his third spell at the club - midway through the year on a deal until the end of the campaign.
Next year will mark 20 years since Jones-Buchanan made his debut for Leeds against Wakefield Trinity Wildcats on May 7, 1999.
He has played 412 times for Leeds, is 13th on the club’s all-time list of appearances and was player of the year runner-up in 2018.
The former England man has confirmed 2019 will be his last season as a player.
Ablett has made 323 appearances for the club, including a record seven Grand Final wins without defeat.
Briscoe, 24, originally joined Rhinos in 2012 from Hull before spells at Wakefield, Hunslet and Featherstone Rovers.
He was a try scorer in a record-equalling 17 consecutive games for Rovers this year and touched down five times in 10 games for Leeds.
Rhinos director of rugby Kevin Sinfield said: “We are pleased to be able to offer all three players new contracts.
“Jamie and Carl are outstanding clubmen who epitomise the culture and ethos we cherish here at Leeds.
“However, more importantly both players are still able to do a valuable job for the team on the field as well and have earned their new deals.
“There has never been any question over Luke’s ability however his early career was frustrated by injuries.
“He is a great example to any young player who suffers a setback.
“He has continued to work hard, listen to his coaches and give his very best every time he steps on to the field and deserves this contract with the Rhinos.”