The actor, known for films including Withnail And I and Can You Ever Forgive Me?, said Joan Washington, a voice coach, had died on Thursday night.
He did not give a cause of death.
Sharing a video of them dancing together to the song Only You by The Platters on Twitter, he wrote: “ONLY YOU! Joan – Love of my Life & Giver of Life to our daughter Olivia.
“Our hearts are broken with the loss of your Life last night. 35 years married & 38 together.
“To be truly known and seen by you, is your immeasurable gift. Do not forget us, sweet Monkee-mine.”
Grant, 64, also shared a series of broken heart emojis, prompting condolences and messages of support from his followers.
He and Washington married in 1986 and share a daughter, Olivia, and a stepson, Tom, from Washington’s previous relationship.
Washington, from Aberdeen, trained at the Central School of Speech and Drama and had nearly 40 years’ experience in the film industry as a voice and dialect coach, working with stars including Anne Hathaway, Vanessa Redgrave and Emma Stone.
Among her early projects in the mid-80s were Yentl starring Barbra Streisand, Highlander, and The Bounty, featuring Mel Gibson and Sir Anthony Hopkins.
Recent years saw her work on features such as The Witches, where she coached Hathaway for the part of the Grand High Witch, and Yorgos Lanthimos’ black comedy The Favourite, where she worked with Stone.
In a joint interview with the Independent in 2011, Grant recalled them meeting for the first time.
He said: “I met her at the Actors’ Centre in London. She was wearing a boiler suit and had fairly short, cropped hair and Kicker shoes, and was chain-smoking, which I thought was insane.
“She was teaching an accent session with a bunch of actors, and I thought she had the most wonderful voice and huge, sort of monkey eyes. She seemed fairly frenetic.
“I thought that she was really cute and all that. It was December 1982.”
Comedians Sarah Millican, Dawn French and David Baddiel were among those sending messages of support, along with actress Eva Pope.
Pope, who appeared in Coronation Street, wrote: “Oh my heart aches for you….deep condolences Richard. She taught me in that very house..”
French tweeted: “Joan Washington will ALWAYS be remembered by those of us who knew her & were taught by her, as a PHENOMENAL WOMAN. So much love to @RichardEGrant & all the family.”