Katharine, 92, was known for consoling losing Wimbledon finalists, notably a tearful Jana Novotna in 1993, and presented trophies at the championships for many years.
She preferred to be known as Mrs Kent and dropped her HRH style, retreating from royal life to spend more than a decade teaching music in a state primary school in Hull.
Buckingham Palace said in a statement the duchess died peacefully on Thursday at her Kensington Palace home surrounded by her family.
"The King and Queen and all Members of The Royal Family join The Duke of Kent, his children and grandchildren in mourning their loss and remembering fondly The Duchess's life-long devotion to all the organisations with which she was associated, her passion for music and her empathy for young people," the palace said.
The union flag at Buckingham Palace was lowered to half-mast on Friday as a mark of respect and a formal framed announcement will be posted on the railings of the royal residence shortly.
A devout follower of the Roman Catholic faith, the duchess became the first member of the royal family to convert to Catholicism for more than 300 years, doing so in 1994.
The duchess will be expected to have a Catholic funeral, attended by senior royals including the King and Queen, which will be the first Catholic funeral service held for a member of the royal family in modern British history.
The duchess volunteered for the Samaritans, toured countries as a Unicef ambassador and, in 2004, founded the charity Future Talent.
Born Katharine Lucy Mary Worsley on February 22, 1933, she had an aristocratic upbringing as the only daughter of landowner Colonel Sir William Worsley - a baronet - and Lady Worsley of Hovingham Hall, near York.
She married King George V's grandson Edward, Duke of Kent, dubbed Steady Eddie by the royal family for his dependable service, in a grand ceremony in York Minster in 1961 and became known for her elegant fashion style.
The pair have three surviving children, George, Earl of St Andrews, Lady Helen Windsor, and Lord Nicholas Windsor.
Her appearance at Wimbledon to present winners' trophies became a familiar feature of the summer sporting calendar and she was famed for offering support to disappointed runners-up.
She put aside royal formalities in 1993, hugging a tearful Jana Novotna after she lost the ladies' singles final to Steffi Graff.
Katharine later became disenchanted with the Wimbledon authorities when, in 1999, she was refused permission to take the young son of murdered headmaster Philip Lawrence into the royal box.
For many years she led a separate life from the duke but the couple did not divorce.
Elizabeth II gave the duchess permission to drop her HRH style when she stepped away from the public spotlight.
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