Her publicist said the star "died peacefully at her north London home with family" on Wednesday.
Wood's long-time comedy partner Julie Walters said she was "too heart sore to comment - the loss of her is incalculable".
Wood found fame in the 1980s and was best known for her BBC sketch Acorn Antiques and comedy Dinnerladies.
She won five Baftas including two for her one-off ITV drama Housewife, 49.
Wood got her first break on the TV talent show New Faces in 1974, while still a student at Birmingham University.
She established herself as a comedy star with her hit show Victoria Wood: As Seen On TV and went on to create the popular BBC comedies Acorn Antiques and Dinnerladies, featuring her long-time collaborator Julie Walters.
The pair became friends in the 1970s and first worked together on TV in 1981 on the comedy sketch show Wood and Walters.
Acorn Antiques also spawned a musical, which Wood wrote and was directed by Trevor Nunn in 2005.
Entertainer Michael Ball, who was a friend of Wood's and worked with her on the musical That Day We Sang, told the BBC she was a "trailblazer" for other female comedians.
The Radio 2 DJ said: "She gave inspiration to other women because she wasn't having to be sexy and rude, although she was all of those things. She was just brilliant.
"She made it seem to other women, you can do this. You just need to look at the various social media feeds to see the people she's influenced."
Ball said Wood, who he had "idolised" before working with, was "very private" and had not wanted people to know how ill she was.
"To then work with her and become friends and to become so close was a privilege," he added.
Fellow comedians have also been paying tribute, including Jennifer Saunders, who tweeted: "Can't believe Vic has gone. She was truly an inspiration and had so much left to to give and we won't see it. She was so funny."
Sir Lenny Henry said: "I am devastated - this is very, very sad news. Victoria will be sorely missed. Always funny, she worked incredibly hard. A killer stand-up and a fantastic songwriter. My condolences to all her family."
Catherine Tate said: "The news is devastating. It's so shocking and sad. We have lost an incredible talent who was a huge influence and inspiration to so many - a true game changer."
Jenny Eclair tweeted: "All of us women in comedy owe a huge debt of gratitude to Victoria - she paved the way."
John Cleese tweeted: "Shocked by news of Victoria Wood. I worked with her last year and was reminded of just what a superlative performer she was. Only 62!"
Jack Dee tweeted: "I feel privileged to have known and worked with the great Victoria Wood. Unique and truly brilliant."
Ricky Gervais said: "RIP the brilliant Victoria Wood. So innovative, funny and down to earth. This has not been a good year."
Sarah Millican also tweeted her sadness: "So incredibly sad to hear that Victoria Wood has died. A true comedy icon."
BBC entertainment correspondent  Lizo Mzimba
Victoria Wood will be remembered for her talent and her outstanding versatility.
Her keen observational style meant she was able to create memorable characters, routines and songs in the world of comedy and beyond - she applied the same skills to her Bafta-winning drama Housewife, 49.
Audiences related to her wit and warmth, and particularly her ability to find the humour in the most ordinary situations.
Most of all they recognised she was one of the most gifted entertainment figures of her generation. Comedian, writer, singer, actor. She could do it all.
Wood appeared on BBC One's That's Life! in 1976, which brought her into millions of homes on a frequent basis.
The show's presenter and Childline founder Dame Esther Rantzen was also among those who paid tribute.
"I think she is one of our greatest comic writers and performers, but she could also deal with serious issues as well, and she's a huge loss," Dame Esther said.
"She did a one-woman show for Childline where I felt hugely privileged to interview her and she told the story of her life with such wit. She just held us all enchanted for a whole evening."
Harry Potter author JK Rowling retweeted a link to a video of Wood performing one of her comedy songs, Reincarnation Song, saying: "Watch and weep. 62 is far too young. RIP Victoria Wood."
BBC director general Tony Hall said: "Victoria Wood was a woman with a stunning array of talents - a comedian, singer, songwriter, actress and director. People identified with her warmth and great charm.
"She brought people from all walks of life together and made them laugh and cry. She will be greatly missed and our thoughts are with her friends and family."
British Prime Minister David Cameron paid tribute, saying: "Victoria Wood was a national treasure loved by millions. My thoughts are with her family."
Wood also broke records with her stand-up tours, where her live comedy was interspersed with original comedy songs performed at the piano, and won her British Comedy Awards in 1990 and 2001.
She performed a record-breaking 15 nights at the Royal Albert Hall in 1993 as part of a six-month tour of the country - something her brother Chris was very proud of.
"The best thing for me was when she played the Albert Hall," he told the BBC. "I was very used to seeing her do her stand up in smaller halls and theatres but for her to be able to come on stage on her own and entertain, control and use an audience of 15,000 was absolutely superb.
"She had a great struggle early on and had to work hard for years to break through. It was her single-minded drive to succeed in the business that pulled her through."
Wood did a lot of charity work and visited Ethiopia in 1990 and Zimbabwe in 1998 for Comic Relief.
She also won the Great Comic Relief Bake Off in 2015.
The star also made documentaries including Victoria's Empire about the British Empire and Victoria Wood's Big Fat Documentary about the diet industry.
She was awarded an OBE in 1997 and awarded a CBE in 2008.
Wood married magician Geoffrey Durham, known as the Great Soprendo, in 1980 - and divorced in 2002.
She is survived by her two children, Grace and Henry.

Sum: Comedian, singer and writer Victoria Wood has died after "a short but brave" battle with cancer aged 62.


The victims' bodies were found in a grave outside the city after family members organised large street protests.
Prosecutors said the crime was a reprisal for the murder of a drug trafficker a few days before.
Three other men received similar sentences earlier this year.
The young men and women ranging between 16 and 34 years of age were reported missing in the days after the group went to a bar called Heaven in May 2013.
Surveillance footage showed some of them being led to cars outside the after-hours bar.
Their bodies were discovered in a grave covered with lime, cement and asbestos on the outskirts of Mexico City.
All of them came from the poor Tepito neighbourhood of Mexico City.
The case shocked people in Mexico City, which at the time had appeared to be less exposed to the overt drug violence affecting other areas of the country.

Sum:
A judge in Mexico City has sentenced two men to 520 years in prison for the kidnap and murder of 13 young people at a bar in the city two years ago.