Williams, 63, known for films such as Good Morning Vietnam and Mrs Doubtfire, had been grappling with severe depression recently, according to his press representative.
Police said tonight he had been found, clothed and unconscious, hanging in his bedroom.
In a statement his wife Susan Schneider said: "This morning, I lost my husband and my best friend, while the world lost one of its most beloved artists and beautiful human beings.
"I am utterly heartbroken. On behalf of Robin's family, we are asking for privacy during our time of profound grief. As he is remembered, it is our hope the focus will not be on Robin's death, but on the countless moments of joy and laughter he gave to millions."
Mara Buxbaum, Williams' press representative, said: "Robin Williams passed away this morning. He has been battling severe depression of late.
"This is a tragic and sudden loss. The family respectfully asks for their privacy as they grieve during this very difficult time."
Police were called to Williams' home in the town of Tiburon at around 11.55am local time yesterday (7.55pm BST) after receiving a call that a man was unconscious and not breathing.
Emergency services, including police and firefighters, arrived on the scene five minutes later, but the actor was pronounced dead at 12.02pm, the sheriff's office said.
It added that Williams was last seen alive at his home, where he lived with Ms Schneider, at around 10pm on Sunday.
Williams shot to fame in the late 1970s as an alien in the American TV comedy series Mork And Mindy. But he won his Academy Award in a serious role, as the therapist in Good Will Hunting.
Stars spoke of their shock at Williams' death.
Comedian and actor Billy Connolly, a close friend of Williams, said: "Robin was both my friend and my hero, a unique talent and a kind and generous man; the world will be a much poorer place without him."
Williams was a regular guest at Connolly's Aberdeenshire mansion, Candacraig House, and has previously attended the annual Lonach Gathering with the comic and his family.
Connolly's wife, Pamela Stephenson said: "Robin was one of the most uniquely brilliant and complicated comic artists the world has ever known. As a performer, he navigated a dangerous high wire at every single moment - with no safety net. My husband and I loved and admired him deeply and will miss him in a way that is beyond words."
Steve Martin, who appeared alongside Williams in a 1988 theatre production of Waiting for Godot, said: "I could not be more stunned by the loss of Robin Williams, mensch, great talent, acting partner, genuine soul."
Talk show host Ellen DeGeneres tweeted: "I can't believe the news about Robin Williams. He gave so much to so many people. I'm heartbroken."
Graham Norton, who had interviewed Williams on his BBC chat show, said: "Such sad news. My heart goes out to his family. The great Robin Williams."
Jonathan Ross, who had also interviewed the actor, said: "Robin Williams is gone. I loved him. Incredibly talented performer. Wonderfully sweet and generous person. Heartbreaking news. So sad.
"What a devastating loss."
US president Barack Obama paid tribute to the star in a heartfelt statement.
He said: "Robin Williams was an airman, a doctor, a genie, a nanny, a president, a professor, a bangarang Peter Pan, and everything in between. But he was one of a kind.
"He arrived in our lives as an alien - but he ended up touching every element of the human spirit. He made us laugh. He made us cry. He gave his immeasurable talent freely and generously to those who needed it most - from our troops stationed abroad to the marginalised on our own streets.
"The Obama family offers our condolences to Robin's family, his friends, and everyone who found their voice and their verse thanks to Robin Williams."
Kevin Spacey, who was in the 2009 film Shrink alongside Williams, tweeted: "Robin Williams made the world laugh & think. I will remember & honor that. A great man, artist and friend. I will miss him beyond measure."
Actress Mara Wilson, who played Williams' daughter in Mrs Doubtfire, said: "Very sad, very upset, very glad I did not have to hear about this through Twitter. Probably going to be taking some time off it for a while."
Williams' Good Will Hunting co-star Minnie Driver said: "My heart's broken. Robin was a beautiful, kind soul. Can't bear that he's gone. So incredibly sorry for his family."