A statement posted on social media through her official accounts described her as an “incredible, fierce and loving mother”.
Alley was surrounded by her closest family members, the statement said.
She was best known for her breakout role as Rebecca Howe in the NBC sitcom Cheers from 1987 to 1993, and received both an Emmy Award and a Golden Globe for the role in 1991.
In 2018 she participated in the 22nd series of UK Celebrity Big Brother, in which she finished as runner-up.
“As iconic as she was on screen, she was an even more amazing mother and grandmother,” the statement read.
“To all our friends, far and wide around the world… We are sad to inform you that our incredible, fierce and loving mother has passed away after a battle with cancer, only recently discovered,” the statement read.
“She was surrounded by her closest family and fought with great strength, leaving us with a certainty of her never-ending joy of living and whatever adventures lie ahead.
The family statement went on to the thank the “incredible” team of doctors and nurses at the Moffitt Cancer Centre, in Florida, for their care.
“Our mother’s zest and passion for life, her children, grandchildren and her many animals, not to mention her eternal joy of creating, were unparalleled and leave us inspired to live life to the fullest just as she did,” it continued.
“We thank you for your love and prayers and ask that you respect our privacy at this difficult time.”
As well as her starring role in Cheers, Alley appeared in various films throughout the 1980s and 1990s.
These included the 1989 romantic comedy film Look Who’s Talking, which she starred in alongside John Travolta.
The pair reprised their roles for the film’s two sequels Look Who’s Talking Too (1990) and and Look Who’s Talking Now (1993),
In an online tribute, Travolta said his time with Alley represented “one of the most special relationships I’ve ever had”.
Sharing a picture of the pair together on Instagram, Travolta wrote: “Kirstie was one of the most special relationships I’ve ever had.
“I love you Kirstie. I know we will see each other again.”
Kirstie Alley’s Cheers co-stars Kelsey Grammer and Rhea Perlman led tributes to the late actress.
Grammer, who played the psychiatrist Frasier Crane alongside Alley in Cheers, admitted that he had loved the actress.
“I always believed grief for a public figure is a private matter, but I will say I loved her.”
Perlman said she had “never met anyone remotely like” Alley, adding that she would miss her “very, very much”.
“Kirstie was a unique and wonderful person and friend. Her joy of being was boundless,” Perlman, who played head-waitress Carla Tortelli on the show, said.
“We became friends almost instantly when she joined the cast of Cheers. She loved kids and my kids loved her too.
“We had sleepovers at her house, with treasure hunts that she created. She had massive Halloween and Easter parties, and invited the entire crew of the show, and their families. She wanted everyone to feel included. She loved her children deeply.
“I’ve never met anyone remotely like her. I feel so thankful to have known her. I’m going to miss her very, very much,” the 74-year-old added.
Alley’s other film roles included Star Trek II: The Wrath of Khan (1982), Summer School (1987) Sibling Rivalry (1990),It Takes Two (1995), Village Of The Damned (1995) For Richer or Poorer (1997), Drop Dead Gorgeous (1999).
She won her second Emmy Award in 1994 for the television film David’s Mother and received a further Emmy nomination in 1997 for her work in the crime drama series The Last Don.
She earned further Emmy and Golden Globe nominations as the titular star of Veronica’s closet, which ran from 1997 to 2000, and played a fictionalised version of herself on Fat Actress in 2005.
In 2016, she appeared on the Fox comedy horror series Scream Queens, alongside Jamie Lee Curtis.
Jamie Lee Curtis said the pair had a “mutual respect and connection” when they worked together in more recent years on the comedy horror series Scream Queens.
“I’ve just heard the sad news that Kirstie Alley has died,” the actress wrote, sharing a picture of them together.
“She was a great comic foil in Scream Queens and a beautiful mama bear in her very real life. She helped me buy onesies for my family that year for Christmas.
“We agreed to disagree about some thing but had a mutual respect and connection. Sad news.”
Horror director John Carpenter described Alley as a “delightful actress”.
“Kirstie Alley was a delightful actress I had the pleasure to work with on VILLAGE OF THE DAMNED. She will be missed,” he wrote.
She also made several appearances on reality television, first as a contestant on the 12th season of Dancing with the Stars, where she finished in second place.
British audiences saw Alley in reality TV-mode on the 22nd series of UK Celebrity Big Brother in 2018, with the Kansas-born actor finishing in second place.
During her stint on Big Brother she shared the house with other famous faces including Coronation Street star Ryan Thomas and Dan Osbourne from The Only Way Is Essex, who finished in first and third place respectively.