In a social media post by her children, they said they and Alley had "only recently discovered" she had cancer.
Travolta, who starred alongside Alley in the 1989 rom-com Look Who’s Talking, posted a picture of her on his Instagram saying "Kirstie was one of the most special relationships" he has had.
He added: "I love you Kirstie, I know we will see each other again."
Curtis, who appeared alongside Alley on the black comedy TV show Scream Queens, shared on Instagram: "I've just heard the sad news that Kirstie Alley has died. She was a great comic foil in @tvscreamqueens 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 things but had a mutual respect and connection. Sad news."
Watch: Cheers stars pay tribute to Kirstie Alley, who has died aged 71
British TV presenter and author Britton paid tribute, saying: "Kirstie Alley has died. I worked with her on location and loved her. Every morning we’d be up before anyone else and drink tea and laugh. ‘Hey, Fernando!’
Britton, who worked with Alley on the Channel 4 show Time Crashers, remembered: "We talked about everything, Everything! In a 6 hour taxi ride we pinched a phone and made illicit calls home. The best."
Steve Guttenberg, who starred with Alley in the 1995 rom-com It Takes Two recalled: "Kirstie met me at her house with a tuna sandwich on my favorite bread. ‘Steve, I know all about you and your love of rye bread., eat some tuna.
"We were rehearsing It Takes Two, and every day she amazed me with her generosity of the tangible and with her soulfulness. The world is a bit empty without Kirstie."
Scientologist Alley - who was runner-up in the final series of Celebrity Big Brother in 2018 - was often outspoken on social media, and in 2020 publicly voiced her support for Donald Trump during the US presidential elections.
Author Kimberley Johnson tweeted: "I'm sorry to hear Kirstie Alley has died. She was a Trump supporter and she blocked me, so we were not on the same page politically but that woman was immensely talented, hysterically funny and I'm grateful for her unforgettable contribution to entertainment. RIP."
Cheers star Kelsey Grammer said: “I always believed grief for a public figure is a private matter, but I will say I loved her."
And Cheers co-star Ted Danson told Deadline: "I am so sad and so grateful for all the times she made me laugh. I send my love to her children. As they well know, their mother had a heart of gold. I will miss her."
Fellow Cheers allum Rhea Perlman, who played head-waitress Carla Tortelli, said: "Kirstie was a unique and wonderful person and friend. Her joy of being was boundless.
"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 and I feel so thankful to have known her. I'm going to miss her very, very much."
Seinfeld star Jason Alexander tweeted: "So sorry we have lost Kirstie Alley. She was a gloriously talented & unique actress and I am delighted I was able to work at her side on occasion. She made me feel welcome. And she made me laugh. As she did so many others. My sympathy to her friends and family."
Alley was mother to William and Lillie, who she adopted with ex-husband Parker Stevenson after suffering a miscarriage.
They said in a statement: "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.
"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. As iconic as she was on screen, she was an even more amazing mother and grandmother.
“We are grateful to the incredible team of doctors and nurses at the Moffitt Cancer Center 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. We thank you for your love and prayers and ask that you respect our privacy at this difficult time.”
Alley was born in Wichita, Kansas in 1951, and moved to Los Angeles in 1980, where she started out as an interior designer and began her involvement with the Church of Scientology.
She began her film career in 1982 starring in Star Trek II: The Wrath of Khan and went on to have a long and successful acting career.
Alley is well remembered for role as bar manager Rebecca in hit sitcom Cheers, which she joined in season 6. She also starred in sitcom Veronica’s Closet and played herself in the spoof reality comedy Fat Actress in 2005.
Alley came second on Dancing With the Stars in 2011 and earlier this year, appeared on Season 7 of the US version of The Masked Singer as Baby Mammoth.