The Roman Catholic nun passed away on Boxing Day at a care home close to the Carmelite Monastery in Quidenham, Norfolk, where she lived.
Jonty Claypole, the BBC's director of arts, said: "Sister Wendy had a unique presentation style, a deep knowledge of and passion for the arts.
"She was a hugely popular BBC presenter and will be fondly remembered by us all. We're thinking of her loved ones at this time."
Born in South Africa in 1930 and raised in Scotland, Sister Wendy anchored a series of popular art shows on the BBC, including Sister Wendy's Grand Tour (1994).
Sad news. Sister Wendy Beckett, writer, art critic and presenter of a much loved and popular series of BBC programmes about painting has died.
— David Sillito BBC (@DavidSillitoBBC) December 26, 2018
According to her publicist, Sister Wendy died from a combination of old age and health problems.
Xinran Xue, a close friend, said: "It is very sad news. Many people see her as a religious person and she was far more than that.
Like so many others, I adored #SisterWendyBeckett. Faithful, funny and wise, she revealed for millions the beautiful vocation of the contemplative life and of the Catholic sister. May she rest in peace with the God of beauty, whom she served so well. https://t.co/j8mLEUthTP
— James Martin, SJ (@JamesMartinSJ) December 26, 2018
"It's a huge loss for the art world. She was a brilliant art critic."
Sister Wendy began studying fine art during the 1980s. She used her writings on the subject to raise funds for her convent.
RIP Sister Wendy Beckett: Carmelite nun & brilliant art critic known as "one-take Wendy" in the BBC. https://t.co/5CXGt6CpUD
— Austen Ivereigh (@austeni) December 26, 2018
Having joined the sisterhood at the age of 16, she went on to gain a Congratulatory First Class degree in English literature from Oxford University.
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